Does it have trouble turning on when it hasn't been on for a while, or only after it has shut itself down? Kinda sounds like a heat issue to me but I'm no expert. If it still has trouble turning on even when completely cool, that would probably eliminate that possibility.
The first time I noticed it it would not turn on though it had been off for over a day. However, the last time I turned it on it turned on with some ease. Shut off after maybe 1-2 minutes -- enough time to start google chrome and check reddit very briefly.
Difficult to say. They were definitely working the last time I used it, but it doesn't stay on very long. Might open it up and have a look this weekend.
Sounds like soldering issue somewhere on the motherboard.. have you ever dropped it?
This is kinda what I had in mind as well. If you've had any heat issues, it is possible that a solder point could have gotten weakened or broken, which could explain the power button issues. Could be a pretty simple fix if you can access it and fix the solder, but if it was caused by some sort of overheating you may have to remedy that as well (fix/replace a fan, clean out dust, use an external laptop chiller, etc.)
The laptop was unplugged the night before so it can't be a power surge. Unfortunate if it's a motherboard issue -- I don't have the slightest idea on how to go about fixing that.