Iphont battery meter not working/ push notifications not working/restarting over and over again.

Question

A few weeks ago, I went on a vacation out of the country. I had my iphone 5 on airplane mode for a week. When I got back and turned it off of airplane mode, it started restarting over and over again right after being plugged into the charger. Eventually, it seems to get ahold of itself and stops turning on and off, but the phone battery meter freezes where ever it is at that point. If it has 100% battery, it says it has 100% battery until it dies. If it only has 1% battery, I'll leave it on the charger long enough to have 100%, but it will still say 1%. The battery life itself is not any different. According to the meter, my phone will stay on all day with 1%. Also, after being dead, my phone would turn on immediately after being plugged in (rather than waiting a few minutes to get enough charge as usual). I think this is because since my phone does not know how much battery it has, it thinks it has enough to turn on right away.
I visited the apple store, and they told me to restore it as a new phone. When I did this, it did not work at first. But after a day, my phone battery life meter started working normally again, and it stopped doing the turn on/turn off continuously thing. BUT imessage no longer worked. It said "waiting for activation" for two weeks. Also, push notifications no longer worked. Every time I openned an app, it would notify my that i could not have push notifications and to connect to itunes. Certain apps wouldnt work at all (snap chat, lulu). They would just give the alert about push notifications over and over again.
I wanted imessage back, so I restored my phone as new once again. This time, imessage, push notification, snapchat, ect. started working again. But, it has gone back to not knowing how much battery life it has and restarting over and over again after being turned on from a dead state. WHAT DO I DO?! I don't want to have either of these problems. It's kind of like my phone has a virus.
note: the first time i turned it on after coming back from vacation, it said "no sim card" for a few minutes.

Answer 1

Option 1
Back Up and try rebuild the library: hold down the command and option (or alt) keys while launching iPhoto. Use the resulting dialogue to rebuild. Choose to Repair Database. If that doesn't help, then try again, this time using Rebuild Database.
If that fails:
Option 2
Download iPhoto Library Manager and use its rebuild function. (In Library Manager it's the FIle -> Rebuild command)
This will create an entirely new library. It will then copy (or try to) your photos and all the associated metadata and versions to this new Library, and arrange it as close as it can to what you had in the damaged Library. It does this based on information it finds in the iPhoto sharing mechanism - but that means that things not shared won't be there, so no slideshows, books or calendars, for instance - but it should get all your events, albums and keywords, faces and places back.
Because this process creates an entirely new library and leaves your old one untouched, it is non-destructive, and if you're not happy with the results you can simply return to your old one. 
Regards
TD