It looks like you are lucky. First of all you should get a utility called TinyUmbrella (http://thefirmwareumbrella.blogspot.com) and save your SHSHs which will allow you to install older firmware on the device, in case you manage to save them before Apple stops signing the SHSHs. Next, see http://jailbrea.kr to tell wether you can jailbreak or not. Lastly, find an apropriate guide to jailbreaking your device via http://iclarified.com
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
Currently there is no jailbreak for devices with A5, A5X A6, or A6X chips; Tethered or Untethered. This means we need to be patient for the Chronic Dev Team to complete the jailbreak for the IPhone 4S, 5, IPad 2, 3, 4 and Mini. Do not be fooled by any site that aims to have it available. Most of them charge and may actually be viruses.
Check http://www.redmondpie.com for any updates. They will have realtime updates as to the status of the progress / release.