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.
Apple cannot tell if you've previously been jailbroken if you do a full restore. Apple has a tutorial on how to do this (support article HT201252).
Jailbreaking only affects the software, and restoring a jailbroken device to an official version of the iOS firmware will make your device "not jailbroken." Doing a full restore can be done to any device regardless of whether or not they are currently or have previously been jailbroken.
Doing a full restore will set the device back to factory defaults for that version of iOS. A full restore erases everything including /private/var/ where all your personal data is stored. If you do an update instead of a full restore, data from the jailbreak could be left behind, since an update leaves files from the previous version.
Additional Information
There is one part of an iOS device called the NVRAM that persists even across a full restore. The NVRAM contains environment variables that are needed for the device to boot. If you have gone out of your way to make modifications to your NVRAM, it can reveal that you have been jailbroken since these modifications can only be made from a jailbroken state. If you don't know how to make modifications to your NVRAM or you've never heard of NVRAM, then don't worry, this doesn't apply to you.
Best Answer
Fully fixed, iCleaner also works, but for some reason when I downgraded rocketbootstrap to v6 everything re-installed and worked properly (except for KarenLocalizer stuff, but that was fixed by borrowing libkarenlocalizer.dylib off of a different device)