Vias - drill sizes and pad sizes
The copper around your drill hole is called an "annular ring". Manufacturers normally specify a minimum value for the annular ring, since drill registration is not perfect. If you have insufficient amount of copper around the hole, you may get breaks in the via, as illustrated below.


When choosing the drill size, you consider factors like available board space and required current capacity. Larger diameter vias will carry more current, but I personally prefer to place many smaller vias instead of a few big ones.

I haven't heard of vias being filled with solder [EDIT: looks like this can be done post-production]. There is a technique, however, to fill vias with conductive epoxy and then plate them over. This is normally done when you need to have a via-in-pad for use with high-density BGAs. See here for more info:


http://blog.screamingcircuits.com/2010/06/how-to-fill-a-via.html
http://blog.screamingcircuits.com/via_in_pad.html