Personally I would create one borg repo for every server I backup. But I also use borgbase, which encourages that by default.

About the backup of the datase - use mysqldump and store that. You can also do it like this:

mysqldump [...] | borg create [...] -

The - tells borg to use stdin as the content you want to store.

Lastly there is a pull mode in borg, that way you could run one script on one host to backup all your servers. You would need to run mysqldump over ssh then.