1) The gear wells are neither heated nor pressurised in any aircraft I know of, so although the gear would cool down more quickly if left extended, the end result would be the same: temperature close to the average TAT in the cruise. It would actually warm up quicker in the descent, so the brakes could be at a higher temperature than normal on landing. Anyway, no real effect.
2) A gear-down ferry flight is severely limited by the extra drag. This reduces the aircraft ceiling, increases fuel flow and cuts the range down considerably, possibly by 50% or more and can also reduce the maximum takeoff weight if the gear cannot be retracted following an engine failure, so everything is working against you. Fairly unusual to do a long sector with the wheels down - it’s normally a hop to the nearest place where it can be fixed.