Background:
'Jumping' is never actually defined anywhere. Opinions differ over whether jumping is possible only in the 8 basic directions, or whether it can be in any direction similar to LOS or whether it can be between two spaces that have X spaces between them where X is the number of spaces jumped (ie, around corners!)
Example:
1 2 3 4 5 6
A OOOOOO P = pit
B OOGOOO M = monster
C OOOOPO G = grappling monster
D OOMPOO H = hero
E OOOHOO
F OOOOOO

Where can the hero jump using 3 MP? Only C4? (it is the only space in a straight line that actually goes over an obstacle). C3 as well?
('Around' the monster in D4). E6? (perhaps to avoid a potential space trap at E5? Trust me, this can be very important!)
If the hero spend 9MP to jump 3 spaces, can he end in A5? Or is he grappled as he passes through C4? He would be grappled if he jumped in a straight line, not if he jumped by choosing space to space and moved 'over' D4-C5-B5 to A5.
Can he spend 6MP to jump to B5? It is straight line, over an obstacle, through no obstructions, but not in one of the 8 basic directions.

Questions:
Q1. Is jumping restricted to the 8 basic directions? If a figure was entirely surrounded by pits (8 spaces), could it jump into any of the spaces on the other side of the pits (16 spaces), is he limited to only 8 out of those 16 further spaces to land in?
A1-1. A figure can only jump in the 8 basic directions, N, NE, E, SE, S, SW, W and NW in clockwise order. So a figure could not jump over a single space and end in a position N-NW away from where he started.
A1-2. A figure can only jump in any direction like LOS. So a figure could jump over a single space and end in a position N-NW away from where he started.

Q2. If the to Q1 is A1-2, then is a figure's 'jump movement considered to be exactly straight like LOS, or calculated by moving over an adjacent space, stating this is the space I am jumping over and then finishing the jump on a further space adjacent to the 'jumped space?
This matters for impassable instructions (and 'grappled' spaces).
A2-1. Jumping is calculated like LOS except there must be no obstructions to movement in the intervening spaces rather than no obstructions to visibility. This means that to end in a space N-NW from the starting space, both of the spaces N and NW of the starting space must be clear of movement obstructions because the line-of-jump passes through both of these spaces.
A2-2. Jumping is calculated like single space movement. So if there is an impassable obstruction to a figure's adjacent N, the figure can
still jump to N-NW by jumping as NW->N instead.

Q3. Can a figure use the jump rules to jump any (passable) space, or only the spaces that specifically allow jumping?
A3-1. A figure may jump any space that he can pass through if he chooses. Thus a hero may jump over a Giant Mushroom, tree or even an empty space should he choose.
A3-2. Only spaces which specifically allow jumping can be jumped. Thus only pits, mud, lava and ice may be jumped over at teh current time of answering.

Comment: This is complex and wordy, but I think necessary. Hopefully more minds can refine things a little bit

Your example diagram is virtually unreadable, and the text isn't consistent with it at all points. Your lists of possibilities are also not all complete (for example, when you talk about a 3 MP jump, you don't consider the second pit as a possible landing space.

Q1 and Q2 seem to overlook some theoretical possibilities, and leave open questions regarding a few details.

Q3 is misleading, because in addition to spaces that don't mention jumping in the rules, there are also spaces that explicitly state that you cannot jump over them. If you're hoping they'll override that, you should ask about it explicitly; if not, the question should be worded to apply only to terrain that currently has no jumping rules and exclude terrain that already explicitly forbids jumping.

I suspect this isn't going to be particularly clear without actual pictures, but I'd recommend reorganizing the questions something like this:

Q1. What kinds of spaces can be jumped over?
A1-1. Only terrain whose rules explicitly allows it. The list up through the Sea of Blood expansion consists of: pits, mud, lava, and ice.
A1-2. Terrain whose rules explicitly allows it, plus spaces with no special terrain, or terrain that has no effects either affecting or triggered by movement. The list through the Tomb of Ice expansion consists of: pits, mud, lava, ice; normal terrain, boulder ramps, corrupted terrain, fog, altars, giant mushrooms, and summoning circles (I'm not sure what you'd add to the list for SoB)
A1-3. Any terrain can be jumped over unless it blocks movement or explicitly prohibits jumping. Up through the Sea of Blood expansion, the only props that can be moved through but NOT jumped over are scything blades and dart fields.
A1-4. Oops, we forgot to specify jumping rules for a bunch of stuff. The intended list is not in any way guessable from the published rules, but should be: (list)

Q2. Where can you land after jumping?
A2-1. Your landing space must be in a straight line from your starting space, following one of the 8 main compass directions, and crossing only jumpable spaces.
A2-2. You must be able to trace a straight line from the center of your starting space to the center of your landing space, passing through ONLY jumpable spaces. Large figures must be able to trace such a line from EVERY starting space to the CORRESPONDING landing space.
A2-3. You must be able to trace a straight line from the center of your starting space to the center of your landing space without passing through anything that blocks movement. This line can cut across corners of non-jumpable spaces (that don't block movement), but it cannot pass through OPPOSITE sides of any non-jumpable space, or through multiple consecutive non-jumpable spaces. Large figures must be able to trace such a line from EVERY starting space to the CORRESPONDING landing space.
A2-4. Jumping just increases the movement cost and lets you avoid the effects of entering or leaving the spaces you jump over. Trace your movement path exactly as if you were not jumping, except you can only pass through jumpable spaces.

Q3. What spaces are you considered to "pass through" when jumping, e.g. for purposes of being Grappled "in mid-air"?
A3-1. You pass through the spaces along a valid movement path that you choose [where every space along your path touches the line you traced, if A2-2 or A2-3].
A3-2. You pass through every space crossed by the line from the center of your starting space to the center of your landing space.

We really need some diagrams showing the difference between A2-2 and A2-3. Here's a quickie, assuming empty spaces are non-jumpable:

_ _
P P
H _

Hero can jump to either of the top empty spaces under either A2-2 or A2-3.

_ _
P _
H _

Hero can jump to the top-right space under A2-3, but not under A2-2, because the line passes through the empty space in the center right.

_ _
_ _

P _
H _

Hero cannot jump to the topmost spaces under either rule, because the line passes through opposite sides of an empty space (in the second row from the top).

_ _
_ _
H _

Hero cannot jump to the top-right space under either rule, even though the line does not pass through opposite sides of any non-jumpable space, because it touches two non-jumpable spaces (those in the middle row) consecutively.

OK, here's some better diagrams. Q2. Where can you land after jumping?

A2-1. Your landing space must be in a straight line from your starting space, following one of the 8 main compass directions, and crossing only jumpable spaces.

A2-2. You must be able to trace a straight line from the center of your starting space to the center of your landing space, passing through ONLY jumpable spaces. Large figures must be able to trace such a line from EVERY starting space to the CORRESPONDING landing space.

A2-3. You must be able to trace a straight line from the center of your starting space to the center of your landing space without passing through anything that blocks movement. This line can cut across corners of non-jumpable spaces (that don't block movement), but it cannot pass through OPPOSITE sides of any non-jumpable space, or through multiple consecutive non-jumpable spaces. Large figures must be able to trace such a line from EVERY starting space to the CORRESPONDING landing space.

A2-4. Jumping just increases the movement cost and lets you avoid the effects of entering or leaving the spaces you jump over. Trace your movement path exactly as if you were not jumping, except you can only pass through jumpable spaces.

Those could still stand to be cleaned up somewhat. Sorry about the giant unnecessary borders, I just exported directly from TileSystem.

Corbon said:

Have you noted how the rules only allow (specifically) the very same obstacles that big monsters can ignore if they are not completely inside? I dare say this two features (jumpable and ignorable obstacles) are supposed to come together. If a monster cannot ignore a tree (for example) then it shouldn't be possible to jump ober it. Or the other way round.

Also, it is said in the instructions that figures in shallow or deep water can't jump. But is there anything to jump over in the water?

Galvancito1 said:

Well, other water for a start!
The fact that figures in water are explicitly not allowed to jump does sort of imply that jumping is not intended to be strictly limited to those obstacles that have explicitly been allowed to be jumped.

There is a level in RTL with water crossing the map. At the end of the "river" there is a pit. I think the answers A2-3 and A2-4 would brake the intention of that level. I do not remember its name, but I can remember that it is the only level that the heroes can enter in a water space. This level would be stupid as the heroes only would need to jump the pit at the end using A2-3 example. See it and you will understand what I am saying.

gran_orco said:

Still of only minor importance. The heroes will mostly actually be faster to go in the water first rather than go all the way to the far end and then jump. Or they can just fly...

Corbon said:

Trust me. It is NOT the same. The first time that we played it the player could jump without problems. Later we said that it could not be jumped: It was awful for them.

gran_orco said:

If you are talking about jumping the water, then yes, it makes a big difference. If you are talking about jumping the pit only, then you may have had a bad experience, but that does not equate to it being a bad thing necessarily.

Corbon said:

I'm not sure about that. It at least means that if you find something you could jump, you won't be able to as long as you're in the water. Maybe there aren't any pits in the middle of (or adjacent to) any water, but this rule could simply be trying to clarify that you couldn't jump that pit even if you found one in such a position. This rule may also have been a preemptive attempt to anticipate such situation before the dungeon/island levels had been finalized, so perhaps even the writers didn't know at the time if anything jumpable would appear in or near water.

gran_orco said:

Water, at least the kind that appears inside dungeons, is explicitly non-jumpable. Bloodsquids, remember? Antistone appears to have forgotten that one in his list for A1-3, but whichever way the ruling lands, the water spaces can't be jumped over. The quest may allow heroes to enter the water, but I don't recall it allowing them to jump the water. (Picky bloodsquids, apparently.)

Steve-O said:

That is why I said 'imply'.
There are no pits, mud, lava, ice or anything explicitly jumpable adjacent to water. I don't think there is even the possibility of such things adjacent to deep water - IIRC deep water always has shallow water between it and any 'empty' space or land that might have a space trap played on it.
So, IIRC, the only spaces ever adjacent to deep water are other deep/shallow water spaces or ship spaces. Which implies, if jumping needs to be expressly prevented, that jumping over those spaces is possible. Since none of those space are, or can be replaced by, a token which is expressyly jumpable, that in turn implies that jumping over spaces that are not expressly jump-able is possible.

It doesn't outright say it, but it surely does imply it.

Steve-O said:

You could play a Frozen Path trap on an empty land space which could extend back into or adjacent to shallow and/or deep water (even if deep/shallow water was classified as an obstacle, which would prevent the ice marker placed on the water spaces). This will create shallow (and probably deep) water spaces adjacent to ice spaces, which are jumpable in general.

Regarding the level with the water that can be entered, I always found it that the heros used it to their advantage to speed them up, and they only bothered with jumping the pit on their way back across the river.

Hmm about the fly thing that is open to suggestion. That level says as soon as you ENTER a water space your turn ends. We had zyla with flying but still the dungeon level says that once you ENTER a water space you end your turn and flying doesn't get around that ruling. We ended up using zyla go over the pit with flying to get the chest as i recall and kill the skeleton.

And by the way we always played that you can only jump on a straight non diagonal way a pit. Doesn't seem game breaking to me or anyone else so why bother my head with more questions and weird rules? I had never even considered these strange jumping patterns. You might as well let apes start leaping diagonally.