If you can find a battery of about the same size same voltage and the same chemistry (make sure of this!) you could probably solder it into to socket. Or you if it is a battery pack you could remove the casing from the pack and replace the individual cells (you can usually find cells of any size on ebay for a couple of dollars). Please note that messing with batteries is dangerous (lithium cells can catch fire). Also make sure you checked ebay for the proper battery for the unit. You would be amazed at what people have stocks of. hope that helps
Once when I worked on a 4th gen touch I lost this piece and the device had no issues on re-assembly. SO I don't think it is needed, but there is a reason it is there so it is best if you can get it back in place.
You will probably need the buttons replaced. Sadly the 4th gen is a PITA to open and the lock button flex cable is soldered to the motherboard making replacement hard. If your volume buttons still work and you don't want to fix the other buttons, you could try to jailbreak the ipod, then you can install an app that lets you use reconfigure the volume buttons to act as the home and/or lock buttons. hope this helps
Hi I Have done this a few times at it is a huge pain. I have broken a flex cable before. TO do this I usually put the casing of the ipod in a small vise and gently lift the circuit board as far as it will go without damaging the cable, use a small piece of plastic, usually the rear end of a small ipod opening tool to hold in this position. Then I use small tweezers to line up the cable. That is just how I do it their are probably better ways