Earlier you said "And he have a jailbreak so he should have this proplem not me .."
The only thing you can really try now is.
Do a backup
Wipe the iPad clean with a factory restore. I'd suggest getting the firmware direct from Apple and restoring from that file.
Once restored from downloaded Firmware login and download an app and see what happens. Make sure you setup as NEW IPAD not a restore
If you are restoring from a Backup you will be restoring any issues that you have.
A few other suggestions
Download a FREE APP from the App store on the iPad. See if this helps as it validates the password.
Another good guide
Yes, syncing information is shared - if the code is written and executed differently (it is) it will cause problems for you. Have seperate user accounts on your pc and separate libraries. Share your libraries on a home network (not the iTunes homesharing). It might also be a good idea to limit the number of Apps open at any one time on your ipad - double tap the home button, press on any App icon until it wriggles, then use the little red'-' in the corner to close it down. doing this when browsing or playing games will certainly help. if you have icloud backup turned on and also are syncing over cellular data this will also put high demands on your ioS and lead to memory drainage.
In an ideal world it would be better if he had his own PC and any songs etc you want to share you share by using a usb stick to transfer them., scanning for viruses etc when uploading them to your machine.
Apps written for jailbroken devices are specifically written for the way they execute and make various function calls to the iOS - dont use them - it is probably this that caused your problem in the first place.
If you have the time - set up your PC's or user accounts to keep them seperate - compare your Apps list in iTunes libraries. Delete any in your library that came from his. Using the Library menu page in iTunes will allow you to permanently delete them from your user account library - I would also recommend permanently deleting them from the recycle bin if that is where they end up.
Since you did a full factory restore using a fresh download of the iOS everything has been ok you say? But if you continue to share apps and syncs/backups - or syncs icloud etc with each other by sharing authorised devices in your account settings - your problems are almost certain to reappear.
You now seem to have (with Shiftlocked's help on reinstalling your iOS) a working pad again, you have spent a lot of time sorting your problems out and much time gained from others help - it would be a shame to waste all that time would it not.
Best regards and hope you have a happy ipadding from here. Nick