At the end of the turn the character will lose the extra Weapon slot. Which Weapon does the character keep?
[Source: WotC]
If it’s your character, you choose which one the character keeps. The other one is scrapped.
At the end of the turn, does a weapon equipped from the scrap pile that is in the “extra” weapon slot get scrapped or do you choose?
[Source: WotC]
You choose.
If I have a LV Gamma Disruptor Launcher on Shockwave and play a Dual Wield for my turn, giving Shockwave another Weapon slot, and I play another LV Gamma as my upgrade for the turn, then attack. Since 2 LV gammas are allowed in one weapon slot, I presume the one attached by Dual Wield will stay on as well?
[Source: WotC]
Yes.
If I play Dual Wield on a mercenary who has no weapons on him originally, do I have to discard that weapon at the end of turn, or because I still have an open weapon slot can I keep that weapon?
[Source: WotC]
You do not! You can keep it in the original Weapon slot.