You certainly can install MySQL on your DreamCompute instance. The only thing to keep in mind is you will need to open up TCP port 3306 in your security group, so that mysql connections can come in. And of course configure my.cnf or similar config on the server side to listen to the public IP address, via the bind-address param.
As for will there be any issues, there really shouldn't be. The only time when you may see reduced speeds is if the webserver is in our Portland datacenter (US-West) instead of Virginia (US-East), as the database traffic would then have to travel cross-country. If you aren't sure where your systems are located you can check this page in the panel: