It’s just another initial draw, so I don’t see why you wouldn’t.
Also why not? Having to remember which bugs you JUST GOT and which bugs you already had adds a not-fun level of additional bookkeeping and the game has enough of that.
Here I’m assuming that you’re talking about the “sudden death” draw…
3) I don’t believe any one gets a bug chip for the single draw in this case, as the deck is treated more like a die roll, determining the whole conflict, and not specific actions from which the characters “learn” or “recalibrate from”.
4) Spending a bug lets the user/superuser draw two cards instead of one during sudden-death.