Conduction only occurs when there's something close to the object (water) to take the heat away. If nothing's nearby (like in the middle of space), then the heat will radiate away from the source―like light from a torch―until it comes into contact with something else.
Does a moving body have more gravity than a body at rest? My logic is that its kinetic energy would increase, and hence its mass
Yes.
Can you go into more detail? My understanding was that as an object approaches the speed of light its resistance to force will increase but its active gravitational mass will not.