Taskers The Home Improvement Co.

I would advise against using exterior paint for a bathroom.
exterior paint contains chemical additives meant for outside (not enclosed spaces indoors).
can cause chest/respitory issues if used inside.

there are plenty of bathroom specific paints available to use.

If you are not a professional, i would paint the frames before hanging the doors.
this will prevent any staining to the oak.

A damp survey is always a good call when there is a number of potential red flags regards damp.
try to use an independant survey company, rather then a survey & works remedy company as you may get a biased/impartial survey.

You need to use a suitable undercoat & primer to stop the rust bleeding through.
just painting on top is not sufficent to hold the rust stains back.

plenty of stain block primers on the market, x2 coats followed by x2 coats of desired finish.