Does the moon always travel the same path across the sky?

Answer 1

No, it does not. It is explained pretty well in wikipedia's "Lunar standstill" article, which says:

"The Moon also changes in declination, but it does so in only a month, instead of a year for the Sun. So it might go from a declination of +25° to -25° in just two weeks, returning to +25° two weeks later. Thus, in just one month the moon can move from being high in the sky, to low on the horizon, and back again.

"But, unlike the Sun, the maximum and minimum declination reached by the Moon also varies. This is because the orbit of the Moon's revolution about the Earth is inclined by about 5° to the orbit of the Earth's revolution around the Sun, and so the maximum declination of the Moon varies from (23.5°-5°)=18.5° to (23.5°+5°)=28.5°. The effect of this is that at one particular time (the minor lunar standstill), the Moon will change its declination during the month from +18.5° to -18.5°, which is a total movement of 37°. This is not a particularly big change, and may not be very noticeable in the sky. However, 9.3 years later, during the major lunar standstill, the Moon will change its declination during the month from +28.5° to -28.5°, which is a total movement of 57°, and which is enough to take it from high in the sky to low on the horizon in just two weeks."

The reason for the moon's orbit moving between the major and minor standstills is that its orbit is inclined at an angle of about 5° to the ecliptic. Due to precession, the lunar orbit rotates slowly around the ecliptic, taking 18.6 years to complete one cycle