Wheel spacers do not add tread width to your tire. That's ridiculous. And spacers do not allow for more positive or negative camber, it just widens your track for more stability which is why they would use it. And the whole point of drifting is to LOSE grip on the corners and using weight balancing to control your movements. And using wheel spacers void the warranty on any rims you get and put more stress on the wheel bearings by messing up suspension geometry.
What you're trying to do it really popular in Europe. But since you don't live in Europe YOU need to design your new suspension with all the variables. You need to get the wider rims (Typical civic aftermarket is 7.5 so I'd say 8 to 8.5)and figure out what offset you need to have it NOT rub on your wheel wells. Then figure what size tires you need with it.
Will it rub? probably....How can I make it not rub?
You need to firgure out what offset you need to push the rim out slightly so it doesn't rub on the inside or on your new fender flares. You're rim should probably be 8.5 inches wide with something like a 50 or 60 mm offset. You do know this will change suspension geometry and put more stress on other parts?
Will handling be affected if I go with the fronts be smaller than the back?
Slightly more understeer but nothing to worry yourself about.
Will handling be affected if I get wider rims?
You'll have a bigger contact patch so you'll get better traction and handling.
Do these rims need to be custom made?
Not neccesarily but don't expect rock bottom prices. If you get a multi-piece racing wheel, the 8.5 inch width can be coupled with the custom offset you need without making a whole new wheel. Or maybe a rim designed for another car but it won't be easy, they probably won't be 4x100 anymore.