Hello !

A simple question : can Lizard use Tail Whip to make a close-combat attack after a friendly Mole Man used Mind Control to assign a close-combat attack as a free action to him ?
Both time he makes a free close-combat attack, are they the same effect or is Tail Whip a different effect than a standard close-combat attack ?

From the Player's Guide:

When a character is affected by Mind Control, any of its powers previously countered remain countered. Powers and abilities that may only be used once per turn, round, or game may be used accordingly per Mind Controller.

So, Lizard can do a close combat attack as a free action and then Mole Man can MC Lizard to do the same thing, because as far as Mole Man is concerned Lizard hasn't done anything on this turn.

That rule applies here ? Even if Tail Whip has no such limitation of "once per time unit"?
Does that also mean that you can use Mole-Man's MC on your own character to activate Outwit twice ?
Would that ruling on Outwit be any different if Mole Man himself had already used Outwit this turn ?

It applies here as all free actions are powers that may be used once per turn.

You could do what you describe with Mole Man and it wouldn't matter if he had used Outwit, because he is not using his own Outwit when Mind Controlling a friendly to use Outwit.

OK I'm confused about something else now :
Lizard has Charge. If I use Charge and then use the free close-combat action in it to make a regular, no-other-effect-activated close-combat attack with it, can I still use Tail Whip to make another attack as a free action ?
Same question if the guy would be able to use HSS (with the rebreather for example) ? Or any other effect granting free close-combat attack ?

Yes, you can. Tail Whip is a different effect than the free close combat action from Charge or the free close combat action from HSS.