From the map just hit the options/select button (whatever the left one is called on Xbox). You can only have 1 active quest and 1 activity marked at a time, though, and there’s no easy way of sorting by planet.
I don’t think so. Inactive quest markers in the immediate area still show a white quest marker, while the active quest remains blue.
It would be really great if the missions menu let you select a filter by planet/system tab, or just by distance. Anything showing more than a few kilometers would let you know its probably not on this planet.