My 20 amp circuit has some 14 gauge wire in it. Is that OK?
Since your 20A circuit has 14ga wire attached to it, it's already illegal -- adding more 14ga wire to it won't change that situation as far as the code is concerned.  But adding load to it may well start a fire inside your walls and burn your house down, since a 20A breaker or fuse can't properly protect 14ga wire.  You're lucky in that it hasn't overloaded so far, but your new ceiling fan may well be the straw that breaks the camel's back.
To correct this problem, and make both your existing installation and your proposed extension legal and safe, you should change the breaker on that circuit to 15A.