If I delete my Gmail address from my Google account (with this method), then could I register a new one? Google support pages don't say anything about it, sadly. I think maybe it is possible because you can register a Google account first and add a Gmail address later, but maybe it is allowed only once.
Your Gmail address is your Google account. You can't separate them. If you delete the Gmail mailbox by using the help page you have linked to, you'll not be able to create a new Gmail address and link it to that account.
You'll need to create a new Gmail/Google account.
Yes, I would say so. Your Gmail account is entirely separate to your Google Services account these days. Note though that you won't be able to reuse tour old Gmail name as it would be archived.