How does Beginner's Luck work with Perception in Burning Wheel Revised? If I am using a Perception-based skill with Beginner's Luck, then I roll my Perception stat at a double obstacle. However, Perception rolls are always open-ended. Does that mean that this roll is also open-ended?
Yes. Beginner's Luck tests are considered stat tests, not skill tests, so Perception-based Beginner's Luck tests are open-ended just like any other Perception test in Burning Wheel Revised.
Note that Perception rolls are no longer open-ended in Burning Wheel Gold.
Yes. Perception is always open ended.
Yes, it does count as a stat test.
As a side note: if not routine before doubling, it doesn't count for learning the skill!
Yes it's open ended. If the obstacle is such that it would count as a stat test, it must be successful to count. If routine (and thus against the skill aptitude), it always counts.
I'm pretty sure this is now formally stated in the Adventure Burner chapter on advancement, but it is just a narrow interpretation of how the rules intersect.