Staying connected to iTunes
When I try to connect to iTunes, the connection lasts for only a few seconds before the main screen returns.
Have you tried using a different/new USB-30 pin cable? If that doesn't solve your problem, there is most likely a problem with the 30-pin receptacle in your Touch. This is a fairly old device so it may not be worth repairing.
That said, I have recently sold my old Touch 2/3 iPods that my kids used and honestly, I could have sold them 10 times over they were so popular. I was very surprised to see that.
Look for a shop that does micro-soldering repairs.