Answer to: 3GS turns off at ~60%I have changed many batterys that die at higher percentages than you would think. Best solution is to purchase a battery and follow the guide here on ifixit to install, or find someone who can help you out. This will resolve your issue 99/100 times.
Answer to: iPhone thinks home button is pressed after home button repair attemptI don't think it being off center would matter all that much unless it was totally off. The length of the cable that enters the connector would not allow for it to be in an inoperable area. You have most likely installed the ribbon cable incorrectly into tho connector on the charging port. I have done many repairs and I would replace the cable again, but this time replace the dock connector (charging port) as well.