So me and my brother in law just played the first mission and they only reason I won as the rebels is cause I could use the weapons twice in a turn.

So my question is once you use a weapon as a hero in one activation of a hero does it get exhausted tell the next round or is it just an all the time thing?

No, you only need to declare with which weapon you are going to use.

You can use the same weapon for both attacks (heroes can play an attack for both of their actions)