Have you tried different headphones with the unit? There might be dirt in the headphone jack causing a weak connection. You might try a tightly twisted Q-tip to twist in the jack holding it upside down for any dirt to come out. You can also apply a little tiny bit of high grade isotropic alcohol (70%+) to the Q-tip and rub it in the port a little bit to loosen up stubborn dirt, but let it dry for 30 mins before trying it out again to be safe.
I've found that ordering them from China is cheaper, but the shipping always takes atleast 3-4 weeks. Ordering from a US customer might cost anywhere between 20-50% more, but the shipping should also be within a week. Your call ;-)
I'd say a week to be safe. Some people know how to take them apart, but don't attempt to take it apart if you don't have a replacement digitizer screen. It requires a bit of empirical experience beyond instructions to get a proper feel for it.
yes, having a separate home button to touch the gold spring underneath it's location will do the trick. It's clicking mechanism is only in the button itself and not required to make contact to the spring if you have one loose from the front panel digitizer.
More than likely a bad USB cable. This has been the case for me 90% of the time when it came to this error and I was testing various iPods and usb cables. 2nd possibility - bad dock port on the iPod itself OR bad usb port (try another?) Some 3rd party iPod usb cables are terribly poor quality and some don't work at all. I don't normally recommend the most expensive parts for a lot of things with good reasoning, but this is an exception for sure - I highly recommend the apple-brand iPod USB cable as they have definitely had the lowest failure rate. They also have the fastest detection times with iTunes as well! Try a different iPod USB Cable, preferably a pretty new one. No matter what brand the cable is, some definitely go bad over time or from damage such as moisture.
The color of the battery doesn't matter. What matters is if the battery is compatible. Is it the same shape size as the other battery being replaced? If so, than more than likely it's the right battery! If it's smaller, but has the same connection port it's probably a 3rd generation battery. Funny enough, that will "also" work with your 4th generation. I'm not sure if it provides less playing time, but it probably does. Never the less, they both still power the devices for hours. Good luck! Make sure it fits ok and don't force the case together if it's not wanting to snap back together. I've seen one replacement battery for a 4th generation not fit and it's pretty rare, but still, make sure it fits!