TRS to RCA should work ok, it does for me.
As far as class A goes, that might be nice (if we're talking tubes) but I wouldn't say it's a requirement.
Class A runs hot, it's always on max. And good class A tube amps are really expensive.
The bottom rung of higher-end audio is fine, something like NAD or similar. I like British. 35w/ch minimum. 50w and up even better.
Forget big-box stores and be very wary of mass market equipment specs. Be wary of mass market equipment.
You absolutely have to go and audition any speakers in person. Speakers are out there at all sorts of prices that'll make you go "wow", but there are an awful lot more that won't. Go around to several audio stores with your favourite cd's (no mp3s). Go back several times. There are huge differences in sound between speakers that may have nothing to do with price. This will be immediately obvious to you once you a/b a few in a store.
I really like KEF Q-series speakers.
Then treating your room will only make whatever you buy sound even better.