Slávek would certainly be possible. Slávek can be used for more names though - for those ending on "slav" (Miroslav, Dobroslav, Radoslav etc.) Maybe also Dobroslávek.
Relative frequencies (in Czech first names): Dobroslav 741/5000000 Dobromír 24/5000000 Dobromil 218/5000000
Yes, it can be used in the Czech Republic, though not very often. I think it's more common in Serbia and Croatia, but it's in the name day calendar on this site (it's not in the diminutive section though).