Heater problem 1999 trailblazer

Heater will heat for a bit when warmed up then cools down? when I rev up the engine it will blow warmer air until I let off the gas?

I cleaned out the heater core and some gunk did come back out? I was going to replace the thermostat but the parts store said the that fi the engine tempature get to 210 then it may not be it?

There is a sgueek but not loud comming from the around the water pump but no water leaks? I touch the lines that feed the heater core one does feel warm and the other is not as warm?

When I was looking to see if the was a problem with the heater door closeing and opening I saw the engine get up to 300 or so but as soon as I reved the engine it cooled down to 210 but he air was cool again?

Need an idea as to what the problem may be Heater core, theremostat, or water pump?

Thanks
Raul