How does this work? I can discard up to the number of brass bullets of my shooter +1 for every brass guy.
Yes, that's how that works.
But do I discard along with my opponent?
The rulebook doesn't specify this, but just as with a real poker game, all players do this simultaneously. Note that I don't have a reference to support this. But it's how I've always played.
Do I show the discarded cards?
Again, just as with a real game of poker, you do not have to show your discards, unless you want to - for some reason. Though keep in mind that since your discards go in the discard pile face-up, at least one discard will be viewable.