The only irreversible thing you could do is to upgrade your baseband, eventually making it harder/impossible to unlock. In the past bricking the device was possible, because you were able to reflash the bootloader incorrectly (iPhone 2g) and thus bricking the device.
PwnageTool and Redsn0w are from the same developers team (dev-team) and they are using the same jailbreaking method (as of redsn0w 0.9.6rc9 and PwnageTool 4.3). IMHO both are safe, but PwnageTool gives you more power.
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
The semi-untethered Pangu jailbreak only works on iOS 9.2 to 9.3.3, and no, you cannot upgrade specifically to iOS 9.3.3.
But, to answer your question about what a semi-untethered jailbreak means, taken from the Jailbreak Subreddit,
A normal jailbreak as we know it is called an untethered jailbreak. Back in the day, there were also tethered jailbreaks. Tethered meant that every time you rebooted your iOS device, you were required to re-jailbreak with a computer (or else your device wouldn't even boot!). Untethered meant that the device can jailbreak itself automatically every time during the boot process. Semi-untethered is a mix between the two - it needs to be jailbroken every time you reboot, but it can be done by an app on the device instead of needing to do it with a computer.