Guess I'll first try to disable tweaks one by one, starting with adblock and Siri (not a tweak though :) ). I suppose there could probably be some leftovers from 5.1.1 clashing with the current system. If I do the big reinstall, I'd like start with a complete deletion this time, and setting things up gradually. Where does one start this -- is the reset/delete from the Settings also valid for a jailbroken device?
Your best bet is to download the 6.1.2 iOS (if you don't already have it) and use DFU mode to install it "fresh." Then, you can start all over again with a new jail break. Do note that if you don't put your backup on - you'll REALLY be starting from scratch; it'll be like new. Of course, you'll have to manually put everything back on and reset your Settings. But, sometimes, the effort helps in "cleaning" up.
If you do indeed have a Siriport on your iPad, take special care in removing and clearing out the modded/moved files because restoring can brick your device even in DFU mode, which is why it's important to backup the original system files before you install Siriport.
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Since you say this can happen "even in DFU mode", you can only be referring to a "hard brick" (i.e irrecoverable) - http://en.wikipedia.org/wiki/Brick_(electronics), no?
A Siri port (aside from probably being illegal) does not alter the baseband of a wireless enabled iPad, and in common with the iPhone, a corrupted baseband is the only way to "hard brick" an idevice, notwithstanding a hardware failure AFAIK.
Since DFU is cooked into the bootrom, a DFU restore cannot be effected by anything installed on the device, no matter how corrupted. That's the reason Apple designed it that way... Whilst I detest the use of the term brick when used in the context of a "soft brick" as we take great pains to explain to our members that you cannot brick an iPad, a so called "soft brick" can always be recovered fully via DFU mode unless something serious has passed me by ;)
We would recommend a full restore if your device is being given away / sold. A restore is akin to formatting the disk and installing a fresh copy of IOS. There is no difference whether you use DFU mode or not in terms of the end result, and to be clear there is no way to perform a forensic "wipe" of the entire storage such as one that would be compliant with DoD 5220.22-M AFAIK.
DFU mode is simply used as a failsafe to cater for any corruption or unexpected data stored on the iPad.
IMHO the risk of the next user of your device being in a position to retrieve usable data though forensic examination of the unused raw storage on the device is so low to be almost non-existent.
Well, formatting a disk doesn't do sh*t, pardon my "French"... :) So there's no such tool as e.g. DBAN (Darik's Boot and Nuke) or SDelete equivalent on iOS? Although I do remember reading about file system being encrypted by default on iOS, but I can't remember the details -- not sure if it was related to passcode lock and whether a device without it is encrypted at all (so that may be a little safer than the default Windows/Linux filesystem).
One course of action could be to restore the device and fill it completely with new data via iTunes, then use the "Erase all contents & settings" option (which will succeed this time, since the device is not jailbroken). That will secure wipe all user data, and since you filled the device, it should in theory overwrite anything in the RAW storage left over from the first time. But nothing is guaranteed ;)
Enterprise customers also have to ability to secure wipe devices remotely, but must admit I have never checked whether this is any different to running the option mentioned above (and it would likely have to be unjailbroken first to succeed).
If you are paranoid about such things, you should probably be taking a sledge hammer to your old iPad and not selling it on / passing it on to a 3rd party ;)
Given the fact that 6.1.3 is out now -- if I do the DFU mode procedure tomorrow, will it get into 6.1.3? I want to keep it on 6.1.2 just in case the next owner wants to jailbreak it...
Edit: I suppose I download 6.1.2 from here http://iosdl.com/ although I don't know if this is a valid source, I just found it on Google.