IPhone 4S not syncing with iTunes 10.7 Asking to restore

Question

I have had my iPhone 4S for about two years, and for a long while now it has been unable to sync with iTunes. My iPhone has the latest iOS, btw, and my iTunes is also up to date.
When I plug my phone into my MacBook Pro, iTunes says that it does not recognize the device and would like me to restore it back to factory settings.
I would like to just restore it, but I do not have any backups for it on my computer. I would like to back it up first, but nowhere in iTunes does it give me the option to back it up at all. iTunes only shows the "summary" tab and does not let me see other lists (music, photos, tv shows, videos, etc). My iPhone, however, syncs fine with iPhoto and uploads my photos/videos with no problem.
Recently, the apps on my phone have been crashing. Aside from contacts, messages, and 70 photos as well as 22 videos, I do not have any other data on it. I do not have any music, and the only apps I have are Instagram and Snapchat aside from the apps that come with the phone. Under usage, it says that I have 14GB capacity and only 2.3GB available. I'm not really sure why 11GB of space is occupied because I don't have very much on my phone. But anyway, because my apps keep crashing, I wanted to factory restore my phone and just add back my messages, contacts, and my photos/videos. However, iTunes will not allow me to back up my phone.
I have tried using external apps for backing up (ex: iExplorer), and apps like iExplorer allow you to save files from your phone perfectly fine, but do not allow you to put them back onto your phone. It is just a way of externally saving your stuff onto your computer. (If anyone knows of an app that I can use to actually back up my phone, I would appreciate it! Assuming that iTunes does not come through...)
I was told a while back (when iOS 6 was released) that with iOS 6, my phone would no longer sync with iTunes because it would now be done through iCloud, but the 5GB of space on my iCloud is not enough to back up what is on my phone, so I have avoided using iCloud for a long time. I was also told that my iTunes may not be up to date enough in order to recognize my phone, but my iTunes is updated and still no success.
I have literally tried everything. I have read every forum on the internet (or at least, that's what it feels like!), and I have even gone into the Apple store (to which the lady said that my phone would no longer sync with iTunes, which I don't understand because I know people with iOS 6 and above who have had no issue syncing).
If anyone can provide some insight into my lengthy issue, I would really greatly appreciate it. I do not want to restore or change any settings until I am sure that I have my phone backed up in a way that can be retrieved and placed back.
Thanks!
PS, can I back up my messages and put them back on my phone? I think it's a yes, but I'm just curious.

Answer 1

Unfortunatley 5 GB is not necessarily a lot of space.  What you should do is FIRST go into iCloud and choose what information you want turned on.
Settings >iCloud
Here you should see different Tabs that will let you select what you want to back up on your device.  You have the option to back up the following data on your phone:
Mail
Contacts
Calendars
Reminders
Safari
Notes
Passbook
Photo Stream
Documents & Data
Find My iPhone
If you turn off some of these options you should free up some of your space. 
Once you have backed up your device in iCloud, you can place your device in Recovery Mode in iTunes so it can be recognized. 
To put phone in recovery mode:  Hold down home button and power button at top of phone at the same time.  Once you see the Apple on the device, CONTINUE to hold down the Home button.  Plug your phone into iTunes.  Do not let the Home button go until you see a message in iTunes stating that a device has been detected in recovery mode.