No.... it wouldn't switch speakers by changing things ahead of the crossover. The crossover is the last device between the source signal and the speaker and you can change everything (music source, head unit, amplifier) and the noise will still be there because the crossover is in the speaker line after all of those devices.
You could even change the speakers in the doors and the problem would still be with the left side because it's the crossover that's getting the noise. You have to move the crossover to try and stop the noise.
As for the whiny head unit, that sounds like a poor ground and if the installers didn't address the ground when they installed the new head unit, look for this one to get whacked in three weeks also. Of course, it could also have been a stray rca pigtail (or unused speaker wire) that got its center pin shorted to ground that caused the original HU problem but, like trumpet said, it's hard to shoot trouble through a keyboard with a blindfold on.
In the end, I would have the installing company explain to me why they think the original HU got damaged and I'd ask what they did to prevent that problem from happening in the future. I'd also ask them to address the crossover issue. Either that or, I'd get to reading this website, cut my losses with the dealer and start working on my own stuff. ;-)