Does an iTunes backup to my computer save restrictions settings? If I have restriction settings on my iPhone 7, and I back it up on iTunes after the restrictions are put on, then will the restrictions be there if I reset my iPhone and the restore from the backup?
Yes, Restriction settings are backed up and will be restored.

Note that if you restore to a different iPhone, you’ll need to do an encrypted backup for the restrictions pass code to be backed up and restored.