Looks very much like yet another machine with a faulty GPU, an issue that plagued both 15 and 17 MBP of 2011. If you have a TV, a thunderbolt to Hdmi adapter could confirm it’s nothing else but the GPU, as an external monitor is driven with the discreet card only. Sometimes one can occasionally see the Mac booting by resetting the SMC.
The only viable long term solution would be to exclude the dedicated GPU by forcing the integrated Intel HD3000 to run at all times. There are several solutions around the internet, each with some plus and minuses and difficulty level, some purely software, some purely hardware and hybrid ones too. You’ll have to take some time to explore the various remedies, google for “MBP GPU disable” and youll’get plenty of sources and discussions.