The biggest issue with mobile nuclear power is the cooling requirements.
No, the biggest issue is shielding requirements. If a regular internal combustion engine for a car can reject heat with a water-cooled radiator, so could a nuclear reactor of the same size. The efficiency of both is around the same.
Air cooling nuclear reactors directly is a mess. Even air-cooling the primary coolant loop isn't done in commercial reactors, because the coolant gets activated as it circulates through the core.
No, primary coolant loop can not contaminate the secondary loop. They are separate, that's the whole point. There are gas cooled power reactors around and they work just fine.
In nautical applications, you can use seawater cooling very efficiently. But even then, nuclear surface ships at times have difficulties crossing the equator, because the intake water temperatures are problematic.
No. Nautical reactors (and power reactors that use a body of water for heat sink) are less efficient when the water is warmer but there is nothing preventing them from operating normally.
Most commercial plants either use evaporation cooling or exhaust the warm water directly into a river or the sea. That's impossible in most mobile applications, because there's no body of water in proximity. Alternatively you would have to frequently replenish a water tank - much like steam engines in the past.
No, reactors can be air cooled just fine. Air cooling is less efficient but the reactor power is far less than a nautical or power reactor.
Outside of nautical applications, the only kind of mobile nuclear reactor that I could imagine, would be a vast freight train. A direct train connection between China and Europe might be the only way this could be economically interesting, if it could be made cheaper, faster and more reliable than ships going through Suez or around the Cape of good hope.
But even then it would probably be cheaper to electrify the whole thing and use electrical power, than to have a nuclear locomotive. And the fact that it couldn't be gauge-compatible with any existing train, and it would have to go through some of the most arid parts of the world (Siberia/Mongolia) basically render it just another complete pipe dream.
But that's the thing with nuclear: if you're going to use it, you're instantly limited to gigantic use cases, as the economies of scale are brutal for nuclear propulsion.