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:
<span style="color:#323232;">mysqldump [...] | borg create [...] -
</span>
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.
Edit:
You can use one repo for all servers, make sure to use prefixes, so that you can use –glob-archives with borg prune, if not you might get old backups cleaned out that you want to keep.