I maintain the Virtualmin installer.
I haven't tested on a DO droplet in a while but we've had a few reports of new issues on Ubuntu...they don't seem to keep their packages as stable as RHEL/CentOS (config files change, versions change, etc. in ways that we have to alter the installer to work with). So, I would guess it is changes in Ubuntu rather than changes at DO, though maybe they've altered their master image for Ubuntu 16.04 in ways that cause trouble. We have seen VPS/VM hosts that provide customized packages or configurations that are weird and non-standard in various ways...historically DO hasn't been one of them, I don't think.
I'm currently doing some more work on 18.04 support (which has been a PITA due to the new netplan network configuration system, which required a bunch of new code in Webmin), but will try to spend some time with 16.04, as well. I've been testing on DO droplets more often lately since it seems to be a common installation target for our users.
The issues you're seeing are pretty mild, however...they're configuration not stability or functionality issues. So, do keep that in perspective...I know it can seem like things are spinning out of control when a bunch of things that used to work start behaving weirdly. It's just config files changing in ways we didn't anticipate when we built the installer for 16.04, not a "bug" per se. CentOS sees a lot less of these problems, because CentOS excludes incompatible changes between minor versions.
Sounds like you've sorted the PHP issue. You can also solve it by simply disabling mod_php entirely (this is what I do on all of my servers, but when I talked about removing it from the default installation a while back, I got a lot of pushback...but, I really think no one should be using mod_php anymore, and Virtualmin 7 will not install/configure it by default).
The MySQL password issue we've seen, off and on, for ages, and I still don't know how to fix it. When it happens, you can usually just back up and do it again. i.e. when you get the error about the password not being set or whatever, just click the button for the previous step or whatever. If it's the problem I think it is, Webmin will either let you set the password again or will detect that the password was actually set and things will Just Work. You can also force a password reset after installation in the Webmin MySQL module.
The import issue is new to me. Are you importing within the Webmin MySQL module (clicking "Manage Databases" in Virtualmin uses the Webmin MySQL module)? If so, have you opened a ticket about it (in the github tracker for either Webmin or Virtualmin GPL, though I think this would be happening in Webmin)? We'd need to see an example database dump that triggers the problem. I'm inclined to think PHPMyAdmin is doing something weird, since our export and import are handled using MySQL pretty close to directly, without much munging of the files, but we shouldn't bail without some kind of reasonable error even if something is wrong with the dump.