What car? If FC is a bit higher that means you're not using lightweight rims.
Lowering your car with sport springs will much improve stability. Strut bar though not so much unless you go racing or jumping up and down in rally tracks. Better spend the money for better dampers and anti-roll bars, they will give much more significant improvement. I'd think 215/50/17 should still be ok though, I'm also using the same size for my front.

Can be the tires & rims not well balanced, can be your engine and chassis already cannot cope and creating excessive vibrations. Tires even though cheap but if balanced well and no bulging then should not vibrate. Balancing problems normally happens only at certain speed range and does not happen below or over that speed range. If the vibration just keep increasing as the speed climbing and never goes away afterwards, then most likely not balancing problem.

Sure, why not?? On wheel balancing is with engine off and transmission in neutral anyway.