It sounds, by your description you need a new dock cable.
If you have another cable or one you can borrow from a friend and attach the to the PC to see if that will fix the problem. You really need the original Apple branded cable to be sure.
If you are still having problems, turn off your and try cleaning the dock with some isopropyl alcohol with a bit of micro fiber cloth. Remember to be gentle. Do not use rubbing alcohol. Do not use cotton swabs.
To answer your other question, yes you can Sync you iPad over your local network if iTunes is running on your PC. You will need to Sync your iPad with a cable to set up the Wi-Fi Sync.