Does the major axis of the moon's orbit point in the same direction (with small deviations) throughout the year? or does it migrate along a path so that after a million years it might be 90° away from where it started?
The direction of the semimajor axis of the Moon's orbit actually changes quite quickly. In fact it does a cycle around the ecliptic in approximately 8.85 years. This is known as the apsidal precession. That Wikipedia article also explains the Moon's nodal precession, the precession of the Moon's orbital plane, which has a period around 18.6 years (retrograde).
These precessions have such short periods because of the strong effect of the Sun on the Moon's orbit (in fact, the Sun's gravitational force on the Moon is stronger than the Earth's). And because these cycles are so short and have a significant effect on the timing of eclipses, they have been known to astronomers since ancient times, although of course ancient astronomers didn't have a good model to explain the cycles they'd discovered.
Because of the Sun's influence, and the fact that the Moon is very large relative to its primary (Earth), the Moon's orbit is actually quite difficult to model accurately. It's certainly much harder to predict the lunar motion than the movement of the visible planets. The ancients managed to do an ok job on the Moon with the Ptolemaic system, although in that model the apparent size of the Moon varied by much more than it actually does. I'm sure this was very annoying at the time, but in the long run it was a good thing, because it made it obvious that the Ptolemaic system was not correct, but it took many centuries before a better system was found.
If you want to know more about the complexity of lunar motion, please take a look at Wikipedia's article on lunar theory, which is an excellent introduction to this fascinating topic.
As pm-2ring already commented, the direction of the axis has a period of about 8.85 years.
It's worth noting, though, that in reality the moon's orbit does not correspond very closely to an ellipse, because it is rather heavily perturbed by the Sun.
Relative to this reality, the concept that the moon's orbit around the earth has a 'major axis' results from treating the real orbit approximately as if it were really an ellipse. The approximations can be made in several different ways, and so in principle there can be a number of different approximating ellipses.
In effect the question presupposes the (idealised) rotating mean ellipse which results (in concept) when nearly all of the perturbation effects are treated as if they were averaged out -- except for the mean rate of axial rotation and the current position of the mean axis.
This idea of the mean ellipse is the one that underlies the 'mean elements' given in tables that express the fundamentals of the moon's motion, such as the 1954 Improved Lunar Ephemeris, which was slightly adapted from Brown's theory and 1919 Tables of the Motion of the Moon.
A more recent estimation of the position and rotation of the idealised mean ellipse, derived from lunar laser ranging via the JPL ephemeris DE200 (official basis of the Astronomical Almanac from 1984-2002), is found in J L Simon et al. (1994) "Numerical expressions for precession formulae and mean elements for the Moon and the planets", Astronomy & Astrophysics 282, 663-683.
According to this source, the mean longitude of the moon's perigee in arc-seconds (counted from the mean equinox of date) (see Simon 1994, p.670 at b.3)
$ = 300071".6752 +14648449".0869 t -37".1582 t^2 $ $ -0".044970 t^3 +1".8948*10^-4 t^4 $
at any time defined by t, where t is the time-interval counted from the J2000 epoch of 2000 Jan 1 at 12h TT (= JD 2451545.0), in units of Julian centuries (36525 days) of Terrestrial Time).
The negative quadratic term in the expression above shows that the rate of rotation of the mean axis is very slightly slowing down; in 20 years it now rotates by about 1 minute of arc less than it used to do in 20 years around AD 1600 in the age of Tycho and Kepler.
Besides the mean approximation to the moon's orbital ellipse just described, there is also the (variable) osculating ellipse. The largest periodic variations in the osculating longitude of perigee, relative to the mean perigee, are given in the same 1994 paper by Simon et al. at p.671, Table 4, 5th column.
Last but perhaps not least is the librating-and-rotating ellipse approximation which results when the rotating ellipse is taken to include the effect of the largest of the solar perturbations, now known as the evection. This is described in older textbooks such as Godfray's Elementary Treatise on the Lunar Theory, see esp. pp.69-71 in the 1871 edition here.