It's impossible to know, for sure, what's going on without having a close look and listen to your audio system, as well as the records you describe. That being said, the TT you have is top notch for its price point. I did some serious seat-time with one 3 or 4 years ago and there was no start-up squeak or slipping. On that front, I would make sure the bearing is adequately greased or lubricated, the floating motor is properly positioned and everything is accurately leveled. If the squeaking and/or slipping continues, I would contact Marantz for troubleshooting. Defective motor? Defective belt? Hard to say! When not playing records, I would recommend taking the belt off to maintain its tensile strength and increase lifespan. Start-up squeak wouldn't bother me if everything else works the way it should. Some TT's, including high-end TT's, do that. This one shouldn't. With regard to belt slippage, I'd make sure the side of that acrylic platter is clean and does not have any kind of contaminant on it interfering with the belt's ability to grip. There is a product you can coat the belt with to increase its stickiness, but you shouldn't be getting belt-slippage in the first place.
With regard to the varying conditions of the records you describe, I would say stop buying or getting them from wherever the heck you are getting them from! That's deplorable! Ultrasonic cleaning or any kind of effective record cleaning is, certainly, a great recommendation. Getting a dust cover for your turntable, if you don't already have one, would be a good idea, as well. I know some vinyl heads wouldn't dream of spinning records with a dust cover in place but I'm not one of them. There is always dust in the air and that doesn't belong on records. Using a good quality carbon fiber brush before play would be another good idea, as well as keeping your stylus clean.
There are a thousand other possible reasons for the sound quality variations but it's impossible to know without seeing and hearing your system.