I have a Rheem split central air system. I was working downtown and my wife called complaining of a bad odor and a buzzing noise coming from the room where the inside part of the A/C system is located. When I got home I found that the blower motor in the inside unit was seized up and I assume had been that way for a while just buzzing away (I assume my electric bill will equal the national debt). I took apart the unit to investigate and was able to get the motor to run for a short period by spinning the squirrel cage. But the motor would stop and buzz shortly afterward. The evaporator coil was turning to a block of ice in 7 minutes, I assume, because the fan was not drawing enough air across the coil. It did not matter if I had the sheet metal panel reattached or not. Also, on the outside unit, a few days earlier, the thermostat would trip the blower to come on but the outside unit would not, but you could hear a clicking noise. Again, I took it apart to investigate and found that the relay was what was clicking. If I manually depressed the relay, the outside unit would turn on. Sometimes until I let go, other times it would stay running, I assume because of the thermostat setting. So, under these circumstances, can I assume that I need to replace the blower motor inside, replace the relay on the outside, and possibly replace the thermostat? If so, is this doable DIY and can you ballpark a cost associated with the repairs? Also, it appears that there is either soot from the cooked blower motor or black mold on the inside of the inside unit. What insulation need to be used to replace the stuff that's there?

can I assume that I need to replace the blower motor inside

Sounds like it unless the blower motor capacitor is defective but too much damage might have already happened to the motor. NOTE when you replace the motor also replace the capacitor with the correct one for the new motor.

replace the relay on the outside

That is a possibility but befroe you do make sure you have 24 volts to the relay/contactor. Many times a low voltage problem will cause the relay/contactor to be intermittent.

and possibly replace the thermostat?

Not unless it is proven defective.

If so, is this doable DIY and can you ballpark a cost associated with the
repairs?

Doable depends upon your skill level and only you know that.

Parts cost different amounts in different parts of the country. Best thing to do is get prices for the parts from where you are going to buy them. Then you will know for sure.

What insulation need to be used to replace the stuff that's there

Just about any insulation will work but the problem you will have is finding it thin enough with a aluminum cover. For this you might have to visit a company that manufactures duct work or sells HVAC supplies. You might find a company that will just sell you whet you need otherwise you will be buying a 150.00 roll. BYW it is usually glued on. Good 3M spray glue will work.

Good Day HVAC1000, Replaced the blower motor and that part seems to be working. Not such success on the outside. Replaced the push switch relay and the problem persists. You mentioned the 24 volts that I assume energizes the coil. What is the source of that 24 volts and how do I get the source to produce the proper voltage? Is there a schematic that I can follow?

Is the contactor coil getting 24 volts?
24 volts originates from the unit inside but the 24 volts is controlled by the thermostat for the outside A/C unit. Might be a diagram on the blower door on the inside unit.

Yellow from the thermostat activates the outside unit. The low voltage tie strip will be located in the inside unit.