How to restore Mac Mail messages lost when changing webhost

No, no other options. You need to keep backups on an external drive dedicated to backups. Email you need to keep need to be moved onto your computer outside the email client's folders and onto the backup drive.

You could ask your former hosting company/email service provider if they have an archive of your email account(s) that you could download.

1 - if you manually archive emails than that would be a place where they could be found. But it's not automatic.

2 - Not sure but probably in the Users/Home/Library/Mail folder.

3 - It's just another mailbox, on you Mac, that you can store emails you want to keep in. However, it's not automatic unless you create a rule for specific emails to be moved there. The Archive mailbox will be at the bottom of Mail's sidebar at the bottom:

4 - create mailboxes "On My Mac" and drag the emails to them.

5 - yes, they will be saved on your Mac in the Users/Home/Library/Mail/V8 folder:

What I do I created mailboxes for specific senders and topics and create rules to move the emails that meet the requirements to the appropriate mailbox. I also created a smart mailbox with the criteria "Mail is Unread" so I can initially see all the emails no matter where they are stored. Once read they will remain in their home mailbox but not in the Smart Mailbox.