If you make a note of the battery charge level, then turn off, plug in the charger for, say 45 minutes, then unplug and turn the phone on you’ll see whether the charge level has changed.
For the power button, I don’t know, but taken in conjunction with the charging thing, I’m beginning to wonder if there isn’t something else that is preventing boot. The phone should always boot when connected to charger.
Do you have a SD card (hopefully formatted as “portable” storage)? If so, try removing it. Try removing the SIM too (but not at the same time so you know what effect is due to which action), just to see what happens.