Calculating AZ and ALT from RA and DEC: values are slightly different from Stellarium
Stellarium 0.14 or later has a checkbox under Configuration, Tools, Planetarium options:
Include nutation.
If I uncheck it, the RA and Dec of date for those stars on 2016/1/1
match yours within 0.2 arcsecond.

Some libraries distinguish between astrometric and apparent positions,
aberration
making up to 20 arcseconds difference.