Question 1. An AT-ST attempts to move through an enemy figure's space. The AT-ST moves into the first square entering the enemy figure's space (2 movement points), and continues to move through that enemy figure. Does it cost additional movement points to continue moving through the enemy figure if the AT-ST is already overlapping that enemy?

The rule says the AT-ST must spend one additional movement point to enter a space containing a hostile figure. The AT-ST entered the space with it's first movement. Does continuing through the enemy still count as entering its space?

Example:

[A] = AT-ST

[E] = Enemy

Starting position

[A][A][A]

[A][A][A][E]

Move 1 square entering enemy's space (2 movement points)

[A][A][A]

[A][A][E]

Move 1 square still overlapped (1 or 2 movement points?)

[A][A][A]

[A][E][A]

Question 2 . Same situation, the AT-ST has entered an enemy figure's space. The AT-ST rotates and the AT-ST is still on the enemy figure's space after the rotation. Did this rotation cost an extra movement point?

The AT-ST does not spend extra movement points to enter spaces containing enemy figures or difficult terrain because it contains the "Massive" keyword. Reference page 16 of the RRG:

The Massive keyword is found on some Deployment cards and refers to a series of special rules that apply to figures in that group.

• Figures do not block line of sight to or from a Massive figure.
• During a campaign, Massive figures cannot enter interior spaces.
• Massive figures can enter spaces containing blocking terrain and impassible terrain. They can also move through and end their movement on blocked or impassible edges.

• Massive figures can enter spaces containing hostile figures and/or difficult terrain at no additional movement cost.

• Massive figures cannot enter spaces containing other Massive figures.

• A Massive figure can end its movement in spaces that contain blocking terrain and/or other figures. Any figures in its spaces are pushed into the closest empty space of its controller’s choice (taking impassible terrain into account). The player moving the Massive figure pushes friendly figures first, and then other players push their figures.

• After a Massive figure ends its movement in spaces containing at least one other figure, the Massive figure cannot move any more during this activation.

Note, the only other "Large" figures are the Nexu and the Eweb. The Nexu has the mobile keyword which allows it to move through figures at no additional movement cost.

Gah, how did I miss that?

How about the E-Web then?

Example:

[E] = E-Web

[R] = Rebel

[ E ] = Boldface overlapping

Starting position

[E][E][R][_][_]

Move 1 square entering enemy's space (2 movement points)

[_][E][ E ][_][_]

Move 1 square still overlapped (1 or 2 movement points?)

[_][_][ E ][E][_]

Move 1 square exiting enemy's space (1 movement point)

[_][_][R][E][E]

And, would rotating on an enemy space cost extra movement points?

Yes on all counts on the E-web. Rotation, movement onto and enemy figure on either point, as long as there is overlap, it will require an additional movement point.

If your E-web is that close to an enemy, it should be shooting twice, and making them fear the rainbow, not moving.