How do you clean the wooden frame?

Dust it lightly with a soft cloth. Don't use modern spray polishes because they build up a sticky silicone residue over time. For dirty frames, a cloth barely dampened with mild soapy water works. Dry it immediately. Apply a good beeswax polish once a year.

My grandmother left me one of these with a broken backrest ratchet mechanism and I wonder if fixing it ruins the piece?

Repairing broken functional parts is standard practice. A day bed needs to support weight safely. Leaving a ratchet broken puts undue stress on the remaining joints. Find a good restorer to splice in matching timber or repair the metalwork. Honest repairs keep these items in use and rarely detract from the object.

What fabric suits an antique day bed?

Linen and wool damasks work well for earlier frames. They breathe naturally and sit right with the old timber.

I have a spot in the hallway near a radiator and a large south facing window, is this a safe place to keep it?

Radiators dry out old animal glue and shrink the timber. You will likely see veneers lift and joints open up if it sits too close to a heat source. Direct sunlight through a large window fades textiles fast. It also bleaches mahogany and walnut. You are better off finding an interior wall with stable temperature and softer light.