iPhone 4S will not sync apps with iTunes

[Dfrtbx]

I actually have two problems with my iPhone 4S:

1. Apps won't sync with iTunes
2. No third-party apps on my phone will open. I'll tap one, a black screen will show for a microsecond, as if the app is trying to open, and then it will close again.

I believe, however, that if I solve the first problem, all it takes is a restore to solve the second problem. I can't restore, however, until my phone is backed-up, which iTunes apparently refuses to do.

Attached is a picture of my iTunes, focused on the "Apps" tab. I've made a few notes in the picture.

iTunes is up to date. My iPhone says that it doesn't need to be updated.

The last thing I did with my iPhone (before all the apps stopped working) was plug it into the TV so that the latter could play music off the former's MP3 player.

I've searched the web for a solution, but nobody's problem, as far as I've researched, was exactly the same as mine. I tried a couple "solutions" out, but none of them worked.

Thank you,
Dfrtbx

EDIT: Shoot! I was a little hasty with the post button and forgot to attach the image. Working on it now!

 

[verwon]

No iCloud backup? Have you tried a reboot?

 

[Dfrtbx]

Sorry, I'm a newbie. What's a reboot? I tried the home button + lock button hold thing, to no avail.

I actually haven't played with iCloud at all yet, given that my phone is less than a week old. Playing with it now, however.

 

[Dfrtbx]

I've just backed my phone up on iCloud. Is it safe to restore?

 

[verwon]

I've just backed my phone up on iCloud. Is it safe to restore?

Yeah, that'll give you a full restore of everything the way it was!