Is recovering cheaper than buying new?
For a well-made frame, often, and you keep something built better than most of what is sold now. For flat-pack furniture, usually not. An upholsterer will tell you honestly which yours is.
Who supplies the fabric?
Either. Many upholsterers have books to choose from and buy at trade prices; others will work with fabric you supply. Ask for the quote split into labour and fabric so you can compare.
How much fabric will it need?
More than people expect, and patterned fabric needs extra for matching. Get the meterage from the upholsterer before buying anything yourself.
How long will it take?
Commonly a few weeks, and longer if fabric has to be ordered in. Ask whether they collect and deliver, since a three-seater is not a car trip.