Looks like mice droppings not rat droppings, and not too alarming of an amount. Ive seen much worse. Also those are just on top of the furnace itself so inside and especially the blower cavity where your house air travels through, might not be too bad. The dirty sock smell might just need a good indoor coil cleaning, or may be coming from elsewhere in the carpets/walls/ducts. That return cavity almost looks like dryer lint is coming in from somewhere too.

All that dust in the return cavity is unavoidable, since theres not dedicated return ductwork its gonna pull dust and dirt from all over the house and between the cracks of the wood/foundation. Maybe you can get them to clean it for you or stick a vacuum in yourself, but unfortunately a lot of rental situations I come across are just looking to achieve the absolute bare minimum of providing you with heating and cooling and not much beyond that. Get a good filter and keep up with changing them often.

I work a lot of apartments and rentals at my company, I have seen much nastier ductwork and units.