Is it (legally) possible to develop an open source solution and have a commercial "pro" version of the same product with more features?
The pro version would be based on the same code base as the free one.
If yes, what's the appropriate open source license for the free version?
That depends on who wrote the code... if all the code you are releasing is your own, then you can pretty much do whatever you like, dual/triple/quadruple-license as you see fit. The licenses provide the consumer of the code with rights, they do not take any rights away from the developer.
Where you may want to be careful though is if you plan to take on contributions from the community in the OSS version, because you are not entitled to include those into your commercial version without open-sourcing that wholesale afterwards. Fine line to walk.
And it may open up arguments that you copied stuff to the commercial version that people contributed to the OSS version even if you didn't, so I'd imagine some care needs to be taken with that particular aspect.
(I am not a lawyer... this is not legal advice)