The old radioisotope generators used Seebecke effect, thus units similar to the peltier modules for cooling. However there is new research to use the IR radiation form a source running much hotter and get there energy this way. They hope to get a better efficiency this way. So there are two different concepts that work. AFAIK the ones used in space flight are still the old thermoelectric ones. The IR radiation method is likely not working well on small scale.
The Pu-238 used for the space flights is different from that used in weapons or nuclear power plants. It needs a completely different path of production to get it in sufficient purity. Still restarting production is easier than going to mars, though is needs an active nuclear reactor.
Helium does not freeze from just cooling (it only does it under pressure). Also there is no way the oxygen could get so could it would even liquefy helium. Oxygen is solid well before helium gets liquid. Chances are more the other way around, that too cold helium could cause some of the oxygen to freeze. A known trouble with liquid helium containers is the outlet to freeze shut (e.g. from humidity) and than cause excessive helium pressure up to an explosion.