Can moon set after being up when due north? (pseudo circumpolarity)
Yes - here is a good explanation of the situation. Just as it's possible to have continuous daylight for days or weeks on end, you can have continuous moonlight too.

The Moon's orbit is tilted at (a maximum of) just over 5 degrees to the ecliptic, however this tilt has an oscillatory precession with a period of 18.6 degrees. This precession means that our "lunar arctic circle" will vary in latitude over time.

For example, if the moon is sitting exactly on the ecliptic (latitudinally speaking), it would remain above the horizon until it is at a sufficiently low phase of illumination that its current ecliptic longitude is below the horizon. Of course in reality the moon only crosses the ecliptic twice during each lunar cycle (it's these two points that precess). Stellarium is a cool program to visualize this on if you turn on the ecliptic grids.

One important thing to note is that this phenomenon is dependent on both the time of year and the phase of the moon. In the summer months, new and near-new moons are the ones that can remain visible for days or weeks on end. Conversely, at the North pole, you might not see a full moon at all for a few months!