Vinyl Newbie - So Many Questions!

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.

Contact the vendor or Marantz about the belt squeak if it’s bothersome. It might go away as the player breaks in. Mine sometimes squeaks on startup, I don’t turn it off to change a LP so it’s one squeak per listening session, no big deal.
Your system deserves a much better phono stage than the $150 entry level Zen. I’d look for something in the $1k range, about half the cost of your TT/cart. Something with a decent power supply, no wall wart. I haven’t been paying attention to amps in that range so I can’t recommend any. Some vendors give a trial period, something to consider.
About the record noise, some are just poor quality and some labels are best avoided. A better phono stage can reduce some or maybe most of it.

1. Give it a helping hand. No big deal.

Assume you've ensured there's no oily contamination on the belt, the pulley, or platter. Ensure turntable is level in both planes. What is the turntable mounted on? 

I can’t help with the turntable, but that is a really cheap phono stage. A really high quality phono stage is really critical. My first Phonostage was a really highly rated $200 one, it was junk. Until I got to the $2,000 category it didn’t sound anything but tinny and thin.

While my turntables, cartridges and phono stages have gotten better over the years, my analog rige did not sound great until I got a Audio Research phono stage… like forty years ago. I would poke around for a used Audio Research PH 3 or higher… 3se, 5, 6, 7, 8… the PH8 I think was the most cost amazing phono stage. 

I was reading about your nice table. The review mentioned that the cartridge should be checked for alignment with an alignment template to be certain that it's correct. It also suggested that the cartridge be as forward as possible. If the cartridge is realigned, that may resolve your distortion issue. I'd also reset your ant skate as this could cause distortion as well. I would do all of this prior to calling your dealer or Marantz.

I use an Origin Live Belt for my modified Thorens and it is better sounding than the Thoren belt. As far as belt slippage, if you have around 60% alcohol, I'd clean off the belt and see if that eliminates the slipping. Don't play warped records without a heavy record weight but a heavy weight would probably play hell with your motor. 

I have an ASR Mini Basis Exclusive phono stage which punches well above its weight class. I'd recommend finding something along these lines on the used market. That is if you're unhappy with your current phono amp. The Clearaudio cartridge should be good enough. I believe Clearaudio cartridges are made by Audio Technica.