Your problem could be one of three things: 1) The cable you're using may be faulty. Try a different cable and see if that helps. 2) The battery inside the controller may be faulty as well. However, they're easily replaceable. And the worst situation would be: 3) The USB port on the controller is indeed bad. You may need to get another controller and/or have your current one sent back to Sony for replacement under warranty (if it hasn't expired). I would try 1) first and see if it helps. If you go with solution 2), you may void the controller's warranty and you'll have to solve 3) by purchasing another one.
A fast and easy way to upgrade your iPod's storage is to get an iFlash mSATA adapter and an mSATA SSD of your choice of capacity. I've done this with my iPod Video and now have 128GB of storage available (more to come down the road!)
I have a pair of Famicoms (both the original and an AV Famicom). They both happily accept the power adapter from either a US Sega Master System, or a model 1 Sega Genesis. This PSU has the correct barrel connector size, voltage, and pin orientation.
You may have a bad video cable. Get a replacement one and try it out. If it continues to persist, there may be either a failing component on the board (maybe the VDP?), or the connector itself for the AV cable is breaking or failing.
Have you managed to successfully restore your iPod using iTunes? That screen usually shows up only if your iPod's firmware is damaged in some way or another. A restore will usually get it working again, but if it still shows up even after a restore, your iPod may have another issue that may need higher up help.