Pumping up tyres, I use a track pump which is meant for bikes but I find pumps up any tyre better than a foot pump, I find. Use separate gauge to check pressure is correct.
Floor? no idea, if caravan is unheated then floor will feel cold. I f you think it might be damp then you need to check it for damp.
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Cold floor - it's winter, the floor will be cold. Carpet helps, as does slippers, as the others have said. I wear a pair of closed toe crocs instead of slippers, as I don't have to change them if I need to pop out for anything.
I pump up my tyres with an electric pump, and have a separate gauge to check the pressure as the one on the electric pump is inaccurate. :)
I bought the little gauge from the local car spares shop.
There is some foam sheeting you can get from Screwfix - or Tool Station. It is used to put under wood flooring. It's not dear. Lay it under the carpet - it should help.