2. Yes. Rocket Jump. It causes Omnitron to be moved.
Move is a specific action. Being moved refers to any action that causes the unit to change hexes. I think Bunker specifying it can't move or be moved is a bit redundant, since the latter covers the former, but it's probably there because if it just said, "Can't Be Moved" people might interpret that as only applying to the specific ability move, this way they can't since move is already mentioned. The third ability prevents him from Sprinting, for instance, because that would cause him to be moved.
Some effects move characters without pushing them. The Operative, for example, has a power (I forget off the top of my head what the final name ended up being, sorry) that moves a character without pushing them.
She's the Operative, she's just that good. She can peirce the skin of Legacy and Beacon with a blowgun. This means she can do some pretty impossible sounding things.
Just find it weird that a character that is unable to be pushed by Beacon can be pulled across the map and flipped into a volcano by someone who doesn't have Beacon's strength.
And if she is that strong why can't she move Bunker? If he's on braced onto a building, just pull the building with him.
You were talking about pulling buildings not characters and the fiction behind it, right?
It doesn't have to make sense in 'reality' if it makes sense for game balance purposes but I can come up with something if you want.
Bunker anchors himself with patented Tachyon Tech, it works with the magnetic fields & iron core of the Earth, so trying to pull Bunker is trying to pull the Earth. Omnitron however is made of non-magnetic materials and doesn't make use of that tech (because the Freedom Five have previously beaten him with powerful magnets), The Operative's chain is made of parts of Omnitron and as it has no defence against its own tech it is compelled to follow the 'order' from part of itself to 'move over here'.
Now she's not strong but knows her enemies' tech and limits, does that work for you?