Upgrading a server that's hosting CiviCRM to get to PHP 7
This looks like a collection of issues, not just a single one, even if they all originated from the OS upgrade.


The CiviCRM status page (assuming it loads) does a pretty good job of checking the usual suspects. I don't think your db would be corrupted.
The CiviSTMP error sounds like the crypto issue. I'd guess that your mailouts are being triggered by a cron running a different php (different version, or different packages).
CiviMail drafts issue + civi running slowly, but only sometimes, both sound like mysql issues. My guess is that your upgrade involved some other changes in the network or mysql settings. For erratic issues, I would definitely finger your provider, not CiviCRM.
Unknown db error - you'll need to look at the detailed error log (sure, install my log viewer!).