I know that Crimea and Ukraine come from a Russian word and crime comes from the French, but is there any crossbreeding between the languages in this case?
Not likely. Wikipedia has cited some good research on the name:
Crimea -- look at the section on Name
The name "Crimea" ultimately, via Italian, takes its origin with the name of Qırım (today's Stary Krym) which served as a capital of the Crimean province of the Golden Horde. The name of the capital was extended to the entire peninsula at some point during Ottoman suzerainty.[6] The origin of the name Qırım itself is uncertain. It is mostly explained as:
a corruption of Cimmerium.[7]
a derivation from the Greek Cremni (κÏήμνοι kremnoi "cliffs", mentioned in Herodotus 4.20).
a derivation from the Mongolian appellation[8] kerm designating "wall", which, however, is phonetically incompatible with the original Mongolian literal appellation of the Crimean peninsula Qaram.[9]
a derivation from the Crimean Tatar Turkic appellation Qırım designating "fortress" or "fosse", from the Turkic term qurum/qurimaq ("defense, protection").[9][10]