The abilities are only active on the turn that they are played. So Lorenzo only gets to counter attack during a turn that blue played their support card.
The Green "Run & Gun" card can't make Leon attack, it can only be used to make green marines attack during the action phase. The Red support ability allows Red marines to attack at the end of the event phase, but obviously this can't be played on the same turn as "Full Auto".
There is one event card ("Quick Instincts") which allows a marine to attack during the event phase. If that marine didn't play their attack action this turn, then they definately don't get their special attack abilities. If they did play their attack action, it isn't really clear whether they get to use their ability in the event phase or not. In the english version, some cards (like red "Full Auto") are worded in a way that suggests it only works during the action phase, while other cards (like green "Dead Aim") are worded in a way that suggests they would work for a bonus attack in the event phase. That is my opinion though, it is subtle enough that the wording will suggest different things to different people.
There is only one "Quick Instincts" card, so it won't change the game too much whether you allow abilities or not. I usually play that none of the Attack special abilities work with "Quick Instincts", just to make the game slightly harder.