If I close my Gmail account, will I be able to re open a new Gmail account on my cell phone number, and will the old one be removed from my cell phone number?
Yes. As far as I know they are not account-specific. So you can assign a mobile number to multiple accounts (e.g., I have my personal account and work account both linked up to my cell phone as a recovery option).
Once you've deleted your old account, the cell phone will be disassociated (though it may take a few days).