On http://batchpcb.com/index.php/Faq it says 20mil (~0.5 mm) minimum drill size. 20 mil is actually .508 mm. As I understand it, that refers to the hole size after plating. The drill to achieve that is larger, but you don’t need to think about that.
The 4-layer boards say “13mil (~0.33 mm) minimum drill size” if you really need smaller vias from BatchPCB.
Yes. Most board houses quote finished hole sizes. I’ve heard that Olimex does not, but I don’t live in Europe. In the Eagle DRC, just set the minimum drill to 20mil and leave it be. I tend to make all my vias 20mil.