Q: When Immobilized, can a figure use skills and abilities that allow movement not related to a move action?
A: Yes. Any skill or ability that does not refer to performing a move action may be used while a figure is Immobilized. This includes skills or abilities that remove a figure from the map and place it in another space, skills or abilities that allow a figure to move an amount of spaces based upon its Speed, and skills and abilities by which a figure can gain movement points.
This FAQ answer (which was there in the first FAQ too, IIRC) clearly states that an immobilized figure is still allowed to spend movement points to move, provided he can get them from some ability other than a Move action or spending fatigue.
Prior to the addition of the other question you quoted, this meant that an immobilized figure could continue using MPs already in his pool. But Immobilize also immediately ends your Move action, which debateably removes the remaining MPs that came from said Move action. This means players need to do math to determine if they have any MPs left or not. This is unfavourable, so FFG has added a new ruling that all MPs the figure currently has are lost on Immobilize. No more math, MP = 0.
But the figure can still move if he can gain MPs through an ability which is not explicitly prohibited. Such are the Rules As Written, and these two FAQ answers taken together make it very clear that this is FFG's intent, and not a perversion of rules lawyers. Common sense, being the rational and objective application of known rules, agrees with this.
Thematic sense, being the desire to "role play" the Immobilize effect or to "pretend you are your hero" says that the figure should not be able to move at all if he's "immobilized." If you want to make a house rule about that, you are free to do so. It's not how the game officially works though.
Just for the record, I totally made a house rule about this at my table. =) For us, anything that allows you to move without spending MPs (there are a few "teleportation" type abilities in there) are kosher, but anything related to MPs or Speed is out until the condition is gone.