D&d Heating

This is possible for a pipe to be leaking and can be but very slow if the copper is unprotected and is corroded but we would need to look at this to confirm

Not likely to heat that space no unless more than one radiator was fitted underfloor heating or more than the one radiator would be best for that size of room

Not unless you can be sure these pipes are dead maybe discuss or book a visit with a plumber just to be sure before cutting

If you are fitting a thermostatic shower the anser is yes you can remove the blending valve under the bath