Why do J2000 coordinates change with time in Stellarium?
Apparently, Stellarium's "J2000.0" reports the coordinates of stars in J2000 frame, but at the epoch of the date you specify, instead of reporting the coordinates of the star at epoch 2000.0, which can be misleading to say the least. So the difference in J2000 coordinates that you noticed corresponds to the proper motion of the star. And the star you picked, 61 Cygni, actually has one of the largest proper motion in all of the catalogued stars, with about 4.1"/year in RA and 3.2"/year in declination. Looking at the differences that Stellarium reports in your example, we can notice that it matches perfectly with this proper motion : 31.3" - 28.2" = 3.1" ± 0.1" in declination, for RA: (55.27s - 54.92s) x 15 x cos(declination) = 4.1" ± 0.1". 

If you pick a star with a smaller proper motion in Stellarium, like Deneb (only 2 mas/year for each axis), the J2000 coordinates change much more slowly.

As for the difference between J2000 RA/Dec and apparent (on date) RA/Dec, this is caused by the fact that the J2000 frame's equator/equinox was fixed at the mean position of the equator/equinox in 2000 (mean position means that the small periodic effects of nutation are averaged out). So the apparent coordinates on Jan 1 2000 are slightly different because apparent coordinates take into account nutation N of the rotational axis R around the mean P (precession):