Just to make sure I understand these correctly.

Large figures don't push or displace figures and cannot end their movement in spaces with other figures, right?

Large figures can be pushed.

Massive figures are Large figures.

Massive figures can end their movement in a space with other figures in it. When they do, those figures are pushed. Massive figures cannot push Massive figures.

Besides Massive figures, who else can push in skirmish? (Reposition command card, Onar and Shyla special abilities)

Are there some other good rules about pushing? I know can get pushed any direction.

Correct, Large has nothing to do with Pushing. Massive has only to do with Pushing due to Massive figures being able to end movement on top of other non-Massive figures.

Basically, apart from that, Push happens only when an ability calls for it. In the campaign Murne's Diplomat's Blaster can Push, Diala's Force Throw, Shyla's Mandalorian Whip, and Onar's Rush are Pushes.

As for rules, see the rules reference guide. It lists two other sources of Push.

Some abilities push figures. The player resolving this ability moves the figure the specified number of spaces.
· The figure can be moved in any direction. It does not need to move directly away from the figure using the push ability.
· A pushed figure is not voluntarily exiting its space.
· Pushing a figure requires no movement points.
· If a non-ability effect pushes a figure, such as a mission rule or if a door closes between two spaces occupied by a large figure, the player controlling that figure determines its placement.
· While pushing a large figure, it cannot move diagonally and cannot be rotated.

Theoretically, it's possible for a figure to be Massive but not Large, right? Or is that restricted in the rules somewhere?

It would be possible for a small figure to be Massive, yes.

Most abilities that Push other figures are restricted to small figures, but with Mind Trick Obi-Wan Kenobi can push large figures too. And because it does not specify otherwise, it is not restricted to non-Massive. Obi-Wan can thus Push an AT-ST or a Bantha one space.

It would be possible for a small figure to be Massive, yes.

Most abilities that Push other figures are restricted to small figures, but with Mind Trick Obi-Wan Kenobi can push large figures too. And because it does not specify otherwise, it is not restricted to non-Massive. Obi-Wan can thus Push an AT-ST or a Bantha one space.

Murne's Diplomat Blaster can also push massive figures, right?

Murne's Diplomat Blaster can also push massive figures, right?

Seems so.

Remember when pushing large figures, the orientation of their base isn't allowed to change.