no
Yes, if it's not being used to store a Time Lord's memories. When Chang Lee opend it in the TV movie, it told the time and was ticking.
In the recent series the Doctor's primary 'gadgets' are: * Sonic screwdriver * Psychic paper * Fob watch
A watch chain is called an "Albert" chain and is often called a "fob" as well.
watch
Fob .
A company called Character Options produces reproductions of the Doctor and the Masters fob watches, which are fairly easy to find in comic book shops or other stores that specialize in SF toys. I think there may also be a more-expensive reproduction available elsewhere.
fob
What is my gold watch fob worth from the year 1844? it was presented to third great grandfather from the passengers of the steamboat harkaway.
Nurses fob watches are sold at Fob Watch Shop, LangtonInfo, eBay, DHgate, Fruugo USA, EachBuyer, Eachmall, Gotham Watches, NurseOClock USA, and NurseOClock UK.
Probably from the late 1930's or early 40's. There is no way to tell for sure.
A watch fob.
A watch chain can also be called a fob chain.