>So then I couldn't use the Express Edition in a client-server scenario, where the app is installed on a server somewhere, accessed by n-number of users? What about MySQL? Is that worth looking at?

You can use express for that. It will not have the full performance of the other versions, but it will work. It is very capable even with limited resources.

Lots of companies use MySQL. I personally have only used SQL Server (and Express) and Oracle, so I can't give you any real input on it.