If you have to boot tethered to make your phone work after jailbreaking with Absinthe....then something is really wrong... And you need to restore... Or prepare for more future problems.
How to make a jailbreak with Absinthe on 5.1.1 exceptionally stable:
1) restore...setup as new device (NOT FROM BACKUP...)
2) after restore, go through the initial on screen setup... Just to where you get to the SpringBoard(home screen). Once you see your SpringBoard..DO NOT touch or configure anything
3) Open Absinthe.exe and jailbreak. (you MUST wait til Absinthe tells you it's okay to open Cydia). DO NOT "Erase All Content and Settings"!!!
4) once jailbreak is done....either:
1) restore a backup that was not taken from your device while it was jailbroken. If all you have is backups from your device while it was jailbroken...then don't restore to any of them. This is the "main" cause of problems.
2) don't restore from a backup at all,.. And just sync music, apps, tones, etc. doing this is recommended and WILL yield the most stable jailbreak always.
5) once done syncing apps, music, etc....close iTunes....open Cydia...wait for SpringBoard to reload.
6) once your SpringBoard restarts, open Cydia again...and wait...wait til it completely loads...it will take a little bit if time the first time.
7) once Cydia has loaded completely...tap on the Search tab and search for Mobile Substrate...install it.
8.) once Mobile Substrate is installed....you can now open Cydia, and start installing tweaks and whatnot on your extremely stable jailbroken device.
You risk compromising the stability of any jailbreak when you restore from a backup....especially a backup from the device while it was jailbroken...or from a backup in which your device had some type of problems when you created the backup. .
Restoring to a backup in which when the device was stock is the safest.
When you don't restore from a backup:
If your worried about loosing pictures...well...that's what Dropbox and other free cloud services are for. Learn to use them...backup pics better then iTunes anyway.
If your worried about losing SMS and call logs. There is in easy way to backup and restore them via SSH...it's not hard at all...google it.
Want to backup your Cydia tweaks's settings....
Backup var/mobile/Library/Preferences. I recommend you backup you tweaks' .plist only,....nothing stock.
Game progress....there is a Cydia tweak for that. Google it.
Anything toggled on in Settings> iCloud will be restored to your device even if you don't restore from a backup. That is what iCloud does.