It sounds like the home button still works, which is the only button you need to have iTunes recover the iPod for you. Note that recovering the iPod will wipe all of its data completely.
I tried this with an iPhone which I believe was in much worse shape than this and it worked. In my case my phone never got past the Apple logo and would shut down shortly after. Even a hard-drive restoration company wasn't able to access the data, yet the recovery through iTunes did work, so there is hope.
Apple calls this recovery mode.
All you need to do is press and hold the home button on the iPod while connecting it to the PC. iTunes should detect an iPod in recovery mode and allow you to recover the device (again, this wipes out all data completely).
I'm inclined to think something has corrupted your preference files or application data, as those will be brought back after a restore from backup.
In order to test this theory, you will need to wipe the phone by putting it in DFU mode, then restoring to the latest Apple firmware through iTunes. Then set up your device as a new device. If the problems still occur (unlikely), then I would take it in to an Apple Store and have a Genius look at it.
If, however, it fixes the issue, then you will have to play "hunt and go seek" with the culprit. Restore your backup, jailbreak (but don't install anything from Cydia except OpenSSH so you can access the file system), and root through your "home" folder (/private/var/mobile).
You should try deleting the preferences (plists) first (~/Library/Preferences), then rebooting. Don't worry, iOS will recreate them. If that doesn't fix it, try removing the sqlite database in ~/Library/Twitter, although I have a feeling the preference files should take care of it.
I wish I had an easy answer for you, but it's going to take some study of the file system and tracing back to the package that "infected" your system.
Not everything in Cydia is deemed safe and you may wish to exercise more caution in the future. MobileSubstrate (which is only made possible be reverse engineering) provides the hooks for many deep level APIs that shouldn't be messed with.
Additionally, you may want to hit up jailbreakqa for additional help, as that is the official forum for help on these matters. From my experience, there isn't much help here regarding this process.
Best Answer
This may pull you out of recovery, but it will clean wipe the iPad. Download the Latest ipsw supported by your iPad. Put the iPad into DFU mode. Open iTunes and press the restore button holding the option + Restore if on a mac, or Shift + Restore if on PC and select the downloaded ipsw file.
P.S: Try this for recovery, it is a paid software. I haven't tried this personally, but the software description says that it can recover data from an iDevice stuck in recovery