Yes, our VPS support databases and most of the languages. You could see more features for our VPS from here: https://www.dynadot.com/hosting/vps.html
For the second question, if the domain is connected with the hosting correctly, it should be good. There might be some propagation issue when you move a domain to another hosting.
And please notice that the VPS hosting is an unmanaged service. Users will be expected to manage their own software installations and troubleshoot issues with their VPS on their own.