My ship execute a maneuver that does not overlap, but does touch another ship's base.

I will perform action this turn?

My ship execute a maneuver that does not overlap, but does touch another ship's base.

I will perform action this turn?

However, technically, except for the one case in the rules where two ships have already bumped and both move with the same Straight maneuvers in the same direction, it is virtually impossible for a ship to land perfectly touching another. Mathematically there will essentially always be either a gap or an overlap between them, possibly thinner than a human hair. There's no way to play to that level of accuracy, though, so a lot of groups treat "perfect touching" as an overlap.

What about of your touching an asteroid but haven't actually gone over it?

Would the same apply?

If you are asking if two objects can be physically adjacent to each other then YES they can.

Can to things be "touching" in game terminology (ie. two ships can't attack each other) without having an overlap to set it up? NO! There are ways to keep touching going from one round to the next but setting it up will always require an overlap to have occurred.

When it comes to obstacles you are either touching them or you are not. The general rule of thumb is that if the ship base/template will sit flat on the table without moving anything then there is/was no overlap but if it doesn't sit level then there is an overlap situation.

What about of your touching an asteroid but haven't actually gone over it?

Would the same apply?

An "overlap" is exactly what the word implies. You have to end your manoeuvre with a portion of your ship's base over the top of the asteroid token for it to count as an "overlap". If you end your manoeuvre hard up against it, but still flat on the table, you're merely adjacent to the token, and not overlapping it.

Now I've seen some players wiggle their ships a little when placing close to the tokens, and basically shuffle the token aside a millimetre or two in order to place the ship flat and avoid the overlap. This is obvious cheating, so anytime a ship might be placed in such a position, both players should take extra care to watch for any movement of the manoeuvre template, the asteroid token and the ship when placing the the ship down. I tend to make sure one player is holding the manoeuvre template firmly down, while the other player holds the asteroid token firmly down. Both players should be very careful not to move either template or token.