1) The screws are very easy to strip. Use a magnifying glass to examine the heads and make sure the screw is actually turning. If you do strip the screw, you may still be able to remove it using a 1 mm flat screwdriver. NB the screwdriver sold on the ifixit site worked well for me removing the two screws that hold the case together, and failed utterly removing the screw that holds the battery clip in.
2) clean the pressure contacts on both the new battery and the phone using windex or rubbing alcohol and a q-tip. Make sure not to leave any cotton fiber behind.
3) The battery ifixit sent me had literally zero charge, and as a consequence it looked like the phone wasn't working. The screen was black and would only wake up when plugged in to the wall. This caused me lots of anguish and I ended up putting the phone together and taking it apart again three or four times. But once I let the new battery charge for 30 minutes it came to life and was fine.