Can I change the seat fabric?

Most antique dressing stools have a drop-in seat. You push it up from underneath and the whole pad pops out of the frame. Taking this to an upholsterer is a simple job. You avoid the cost of transporting the wooden frame and there is no risk of damaging the polished timber during the work.

I bought a stool but it sits quite a bit lower than my dining chairs, is this normal or has someone cut the legs down?

Dressing stools sit lower by design. They need to slide right under a kneehole table, which typically has a central drawer that hangs down and reduces the clearance. Dining chairs are built for eating at a standard table height. If a stool matches a dining chair, it will trap your legs against the vanity apron. Check the bottom of the feet. If the turning finishes naturally, the legs are original.

How to clean the woodwork?

Dust the frame with a dry cloth. If the surface is dull, apply a thin coat of clear beeswax polish. Avoid aerosol sprays because they leave a cloudy silicone residue.

The webbing underneath looks like it is starting to sag quite badly in the middle, does the whole thing need taking apart to fix this?

Sagging webbing means the tension has gone. The old tacks eventually rust or pull out of the wooden rails under the weight of daily use. An upholsterer will strip the bottom layer and tension fresh jute webs across the frame. They do not usually need to touch the top cover or the horsehair stuffing if those are still in good condition.