I'm a software engineer for a small company but I dabble in a little bit of everything. I've always loved taking things apart and repairing them because it let's me learn and make things better. I'm a tinkerer and won't ever stop
I don't have an IPhone so I can't tell you what the piece is for. However from the pictures it looks like that piece sits underneath the battery connector. Looking at pictures 1 and 2 on step 5 shows it's placement.
I did some searching and found that with some of the Sony Cybershot's the type of battery also makes a difference. Most sites I read specifically made mention to the cause being a difference between the type of batteries (e.g. Alkaline vs Lead). The common fix was making sure you were using Alkaline batteries. However, this was from searching for just when it was turning off very shortly after turning the camera on so I'm not sure if that will be useful to you as well in the current state you are in.
Does it shake without anything inside of it? If so it sounds like it's a balancing issue and you can try leveling it. If it is level you can also try shifting it slightly while it is running and see if that helps.
I had someone come to me before with this problem and we ended up fixing it by just turning it off and then back on again. The only other thing I can think of if that doesn't fix it may be something wrong with the jack where the headphones make a connection and can produce audio but the ipod can't recognize that the headphones are plugged in.
I'm not sure what you mean exactly by black spots on the lens. If you mean you see black spots in your pictures, then it could be you have a dirty lens. Have you tried cleaning the lens with a cleaning brush?
Foam pads are normally just isolation devices, intended to either act as a cushion or to separate parts that could make contact with each other. I'd replace it with something similar but I'm not sure it's totally necessary.
I realize I'm a couple of months late on this but I don't believe it's bad luck on Matt's fault. I worked for tech support at a college a year back and saw many hp's come in with this same problem. The consensus that myself and the other techs came up with is that it's a design problem on HP's part. I can't recall if the dv9000 is the same way but others I have seen have the hard drive(s) placed in the front where most people rest their palms. The structural support there isn't usually enough to prevent the pressure from that from contacting the hard drive. Given enough time and depending on how much you rest your hands/how much pressure you put there, it could cause contact between the shell of the hard drive and the read/write arm and cause damage. We were able to replicate this with another hard drive and laptop and did end up notifying HP that we were seeing this problem.