Hi zer0angel,
I see that you have some questions about making backups of your apps in iTunes, and that your apps are currently not syncing to iTunes. If these are apps you have purchased through your Apple ID, then here is a method for re-downloading these past purchases:
Download past purchases from iTunes - Apple Support
https://support.apple.com/en-us/HT201272
There also may be some helpful information in the following article that may explain why this isn’t happening as you thought it might:
If you bought the iTunes Store content on your device using multiple Apple ID accounts, you'll need to authorize the computer for each one.
Redownload or transfer your iTunes Store purchases from an iPhone, iPad or iPod to a computer - Apple Support
https://support.apple.com/en-us/HT201267
Thanks for coming to the Apple Support Communities!