I can see it two ways.
1) It says that he'd take no other action.

That said, regardless of whether or not you are able to complete the first action, he would still have been given that action. Since he was given an action, he would receive a token.

2) Since the first action was rendered illegal, it was fully retracted (as you can't give a figure an illegal action. As there's nothing else to do, he takes no action and receives no token.

Personally, I think #2 is more fitting to the rules, but my gut tells me that he'd get the token anyway.

If it ends up being #2, does this mean that Jean Grey could then TK an opposing character adjacent to Groot, then Groot can attack that other character since he hasn't taken a non-free action this turn?

If it is #2, I'd see no reason for that to not be allowed.

Remember at this point the colossal rules allow him to make ranged attacks regardless of adjacency.

I'm disappointed that the rules posse knew of this issue, made a ruling on part of it, but didn't figure the whole thing out.

Groot would still get a token. This as been noted for the following PG.

OK, but does he end up not having acted at all (even if he does get the token) or does the whole mess still count as his action for the turn ?

It's his action for the turn. Groot's been given an action. He tried targeting the Shape Change, the Shape Changer caused that targeting to fail. While the attacker is free to "try something else," they are committed to receiving a token if they have already initiated a non-free action.