Points #2 and #3 perfectly match with a faulty GPU. Point #1 is a bit unusual, but one never knows what a faulty GPU may cause, a bit of testing would be needed to confirm. You can try google “Macbook Pro GPU disable”, there are several tested solutions, software, hardware and hybrid to run the machine with only the Intel HD3000 graphic if that is acceptable for you. Since it doesn’t seem to even boot, you’ll probably need a Linux boot distro in any case to have the Mac boot.
Or you could try “baking” the board, that may buy you some time although it’s not a long term solution. I would avoid throwing money on a GPU replacement, unless with a well reputed and honest lab.