My iPhone 4s on the latest update freezes even after a full restore.. is it gone?

Question

     At first, my iPhone 4s completely froze and I was unable to turn it on (even after holding power/home button--the apple logo would come on and then it would just turn off).  I wasn't able to get it into restore or DFU mode.  After some troubleshooting, I decided to order a new battery.  I replaced the battery and was then able to get it to restore, however--after I restore it and it comes time to restart the phone, it gets stuck and doesn't turn on.  If I unplug the phone and then hold down the power and home buttons for about 20 seconds, it restarts and shows the apple logo and shows the initial lock screen.  If I let it scroll through the different languages for the "slide to unlock" text, it gets to what looks like japanese or chinese and then freezes.  I then have to hold power+home again to restart it.  If I slide it immediately when it shows up, it initially froze right away--but after another restart, it worked.  I then am able to set it up and get to the wall of apps screen.  From there, the phone worked for about half a day and then it froze up again (black screen, couldn't wake it up, stuck at apple logo and then just turns off).
     I then was able to restore it AGAIN (on my own personal computer with a fresh install of itunes, I'm doing this for a friend) but the same symptoms persist--it will freeze if I don't unlock it right away, it freezes once when I unlock it right away, and after another restart, it gets through setup and to the wall of apps.  The first time I set it to restore from iCloud and re-installed all of her apps.  This time, I signed in with her Apple ID and enabled iCloud but did not download any of her old apps.  When I go to the apple store, under featured and top apps its just a blank white screen but when I search for an app it does show results.  As of right now (30 minutes after restore) it IS working, but it worries me that the same bugs happened after a restore.  I feel it's only a matter of hours before the phone freezes up again and requires a restore.
She does not have apple care and even though she loved this device, she isn't going to buy another iPhone because of all the problems shes had with this one (it's just over a year old).  Help keep an iPhone fan!  Do these problems sound familiar to anyone?  I'm an android user myself and I love rooting/flashing new ROMS, so if jailbreaking her phone or doing some kind of super-restore with 3rd party programs might help, I'm all for it--I just wanted to get some opinions on if this phone is even worth working on more than I already have since the issues remain even after multiple full restores.
Any help would be appreciated!  Sorry for the wall of text.
-Adam (on behalf of Kathleen Morton)

Answer 1

After some troubleshooting, I decided to order a new battery.  I replaced the battery and was then able to get it to restore, however--after I restore it and it comes time to restart the phone, it gets stuck and doesn't turn on.
Sorry, your repair ruined the iPhone.
She does not have apple care and even though she loved this device, she isn't going to buy another iPhone because of all the problems shes had with this one (it's just over a year old).
After you ruined the iPhone with DIY, Apple won't touch it anymore. You should have asked before all these had happened.
I'm an android user myself and I love rooting/flashing new ROMS, so if jailbreaking her phone or doing some kind of super-restore with 3rd party programs might help, I'm all for it-
Yeah, all Android user can think of is Jailbreaking, read this: http://support.apple.com/kb/HT3743