Sorry for your trouble. That does sound as though the logic board is dead, which is usually the case when you see a set of horrible symptoms like this. Buying and replacing a logic board is a non-trivial task. Even finding a logic board is fairly difficult, although if you do there's a good bet it'll be less than $1500.
You could try removing one or the other RAM chips that are in your machine and booting, to see if it's bad RAM.
My advice, based on years of tinkering with Macs: you bought a used Mac 2 1/2 years ago, and unfortunately this can often represent the full lifespan of a Mac. If it were me, I would not invest my time in trying to replace a logic board, and instead would invest in a new machine.
Good luck. Let us know what you do.