DoesCode permit connecting receptacles to a 20ampere branch circuit using short lengths of 14AWG conductors as the pigtail between the receptacle terminals and the 12AWG branch-circuit conductors?

No. This practice is a violation of Article 210.19 (A) of the NEC. "Conductors of branch circuits supplying more than one receptacle for cord-and-plug-connected portable loads shall have an ampacity of not less than the rating of the branch circuit."