Rallyes-New vs OEM

On what car? I don't think I have heard of any issues with 6" on the front along as you don't use too tall of a tire. 7" wide with 4-4.5" backspace work on Darts and Scamps but they are very close. I have only known acoupke guts that use 8" on the rear of those cars but they did have to do some fender lip trimming and rolling.
Trim rings coming off aren't an issue as long as you don't drive it sideways like you stole it.

Aftermarket rings and center caps should work fine on stock wheels.
It's a 67 Barracuda. Car has 8" US Indy mags on rear with 235/60/14. 6" US Indies on front 195/70/14.

All of these have a 3-3/8" backset.