Q: If I sync my IPod and transfer it to a new IPod/Phone will it log me out of my apps

See:

Transfer content from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support

If you restore from iCloud backup the apps will be automatically downloaded. If you restore from iTunes backup the apps and music have to be in the iTunes library since synced media like apps and music are not included in the backup of the iOS device that iTunes makes.

You can redownload most iTunes purchases by: 

Downloading past purchases from the App Store, iBookstore, and iTunes Store        

https://support.apple.com/en-us/HT201272

Note that unless you use an encrypted backup to restore from you will have to reenter passwords for items like mail accounts