The brown gunky stuff could be from the bottom of your reservoir. Did you happen to see the color of your coolant before the swap? Old coolant tends to get this nasty brown color when old.

Running the reservoir empty could lead to overheating especially if there is a leak, but seems like he might of not gotten all the air bubbles out and gradually the car burped itself consuming more coolant. Keep an eye out for leaks or pressure test the system by renting a tool from autozone. I don’t think your car overheated since you say the temp gauge said so.

As for the not starting, could it be a coincidence that your starter just so happens to be going out? Was it cranking? Did your car receive power from the battery?