These instructions are inaccurate: they say the screws are both 1.5mm, but the left-hand one is longer than the right-hand one. If you try and replace them swapped, the left one won't engage.
It's been a month now, and I've been using my phone with no problems at all. Battery life is great, it has all sorts of reception, and there are no unusual sounds as the phone is shaken or put under pressure. Whatever that piece is, I feel like it's not important enough to put back in.
Once my screen was removed, a piece of rubber fell out of the phone onto my work surface. I sweated and looked for half an hour, at the frame in my hand and at all the photos in the guide, for a place that it might fit. I eventually gave up, and my phone has been working fine without it, but where is this piece supposed to go?
It is not strictly necessary. As mentioned above, removing the battery is the only way to be sure that no parts of the logic board are electrified. It is very easy to replace the hard drive without removing the battery, but it is safer to remove the battery first.