The clips are easy to remove. there is a slot in the bottom of them for a flat head screw driver. Just turn them until the clip is loose.
If you have the room you could pull the sink attached to the counter. Either way you need to disconnect all the plumbing.
Are both paints new or just the green?
If just the green is new, try using Goof-off and a rag, it will break down the new paint first.
or you may have to repaint the white, you will have to do that if you sand it.
The thermostat is not repairable but they are replaceable, on gas or electric.
Find a dealer in your area, make sure you take the model and serial #'s with you.
I do not think you will find anything with that cut to it, you will have to make it with a router or use a peice of square stock and fill the gap with wood putty.
Just a word of caution, if you are going to HD make sure it is cellular vinyl, they may carry some cheap foam stuff. I would use a real lumber yard if you can they will let you know what thier products are made of.
First thing is go to the outside unit and see if the fan is running, if not check the breakers, still not Call an A/C repairman, you need special tools to check any further.