iTunes sync question

While you can manage what songs sync to your iPhone by checking/unchecking, it's really not the best way to manage syncing. For example, I much prefer to manage a playlist in iTunes and just sync the iPhone to that. You can then leave EVERYTHING in your iTunes library checked -- iTunes will only sync to what's inside that playlist.

In any case, if you want to continue doing what you're doing, you need to leave the songs checked that you want to remain on the iPhone. If you uncheck them, they'll be removed from the iPhone next time you sync. So always keep songs checked if you want them on your iPhone.

For reference, here's an article on using iTunes to sync media to your iPhone:

https://support.apple.com/en-us/HT201253

Regarding your backup, you can choose to backup to iCloud to avoid backing up to iTunes each time you sync your iPhone. You do need to have an iCloud account, but you don't have to use it for email, etc. But if you want to continue backing up to iTunes, chances are it takes a long time due to you having a huge photo library on your iPhone (i.e. in Camera Roll) since photos are part of your backup. Not much you can do about that but transfer all/most of your photos to your computer and keep only what you want on your iPhone, or delete the ones you don't want anymore from your iPhone (but you should transfer photos to your computer for safekeeping). Here is info on backing up your iPhone:

https://support.apple.com/en-us/HT203977