Currently I have 2 Science Officers, 1 Engineering Officer, and 0 Tactical.
Get yourself a tactical officer. You're wasting an open bridge station by not having one.
I am being offered an Engineering officer, I guess because I've reached a certain level.
Problem is, I don't think I need him.
Every few levels you'll get offered bridge officers. You can accept them from the mission, they won't join your crew until you comission them from the assignments tab. You can also just have them train your own BO with one of his skills and depart if you like. Or, you can trade the candidate with another player. You cannot at this point trade BO's already under your command though.
1) Can Bridge officers be swapped for others? If so, does it cost anything extra?
2) Can I trade bridge officers I've already used?
As mentioned above, you can trade your cadidates, but not ones under your command. You can requisition new ones at earth starbase too at any time.
3) Is it common to end up with "too many" bridge officers?" Like, more than the slots on your ship? I'm worried about not using them properly.
Ship tiers 2 3 and 4 all have only 4 stations. Tier 5 ships have 5 stations. You start out with the ability to have 4 officers under your command. I think you gain slots for 2 more each promotion, so that would mean you'd have a total of 12 by the time you're an Admiral, so far more than the number of stations you have.
4) Is it possible to ever choose the race of your Bridge Officers? Or just have to take what's given?
If you go to the BO requisition officer in Earth starbase you get to choose among a variety of different BOs of different races... though you are limited to what's available there which is usually a pretty decent variety.