"After removing the bottom cover and disconnecting the battery, the laptop will power back up and run and continue to run if I reconnect the battery and disconnect the charger. I can even put the laptop to sleep by closing the cover, and it will wake normally." Use this technique to power on the machine so you can fully test the functionality of the top case. To test the power button try to force the machine to power off by holding the power button for 10 seconds (also, if the charging light changed when you performed the SMC reset the power button is working). I strongly suspect the battery has an issue since you are only experiencing problems when the battery is connected. Batteries for this model are relatively cheap and easy to install. I would recommend installing a new, known-good battery to see if this resolves the issue.