We played our first game last week and came across the situation wherein the rebels successfully attacked and held a system loyal to the empire. Unfortunately, they were not able to clear the Imperial Star Destroyer out of the system, which left them occupying an imperial planet while the star destroyer remained.

If I understand correctly, the following happens:

1. Loyalty remains Imperial. - assumed because I found no mention of the loyalty changing, unlike what happens when Imperial ground units occupy a system loyal to the rebellion.

2. The Imperial player cannot build new units from the build icons on the planet, nor can he place them on the occupied planet.

1. Loyalty remains Imperial. - assumed because I found no mention of the loyalty changing, unlike what happens when Imperial ground units occupy a system loyal to the rebellion.

Correct in this instance, but when the Imperial subjugates a planet, it does not change its loyalty. The system is subjugated and either Rebel Loyal or Neutral.

2. The Imperial player cannot build new units from the build icons on the planet, nor can he place them on the occupied planet.

Correct as well. (He also can't place space units in the system, but I think that's what you meant)

Subjugation does not make a system loyal, so if you subjugate a neutral system it's still neutral for the purposes of card abilities it's an imperial system but not a imperial loyal system

2. The Imperial player cannot build new units from the build icons on the planet, nor can he place them on the occupied planet.

Correct as well. (He also can't place space units in the system, but I think that's what you meant)

Yup. That's what I meant.

Loyalty is unaffected by invasions. The only potential loyalty effect is through subjugation - which does not change loyalty, but only adds a subjugation marker over the existing loyalty (rebel or neutral). As long as a rebel or neutral system contains imperial ground units, it is subjugated. Space units do not subjugate systems, but they blockade them - preventing material support coming or going from the system.