Heater doesn't work

Ok so we need to see what the recirculation system is doing. If you start the car and turn on the heater and switch from inside to outside air do you hear the door move? If not then I would like you to call around to your local parts stores and find one that will do a free scan and have a scan ran to see if the heater is giving off any codes. Let me know what you find.

So when the engine warms up is both heater hoses getting too hot to hold?

Which engine do you have?Also when the engine is at normal operating temperature is both heater hoses too hot to hold going into the heater core?

The sludge is a result of lack of maintenance on the cooling system, it has clogged up the radiator and the heater core, and yes, I believe you have damaged the head gaskets.

I do not reccomend you try this yourself. It is very involved job.

Roy

Could be automatic transmission fluid in the coolant due to a leaking cooler in the radiator. Check the condition of that fluid. The cooling system will remain under higher pressure right after the hot engine is stopped so coolant will sneak into the transmission fluid.

Happy to hear about the transmission but sorry for all the other problems. Don't be too hard on the dealer. People trade cars in without telling them about existing problems in the misguided thinking they will get more dollars for their trade. The dealers also often buy cars from auctions and they don't know about existing problems unless they show up on test drives which aren't very long. That's why they often give 30 day 50 / 50 warranties. They will take care of the things that show up later. They don't make any money on those repairs but they don't lose money either. It's a better deal for you than buying a car "as is".