Daniel,
My experience is limited because of the short time with the BB UL, six days/nights out. So far I have had it out on one cold drizzly day / night. I pitched the silnylon tarp high, about 4.5 ft at the point where I used the BB. I had no problems with the tarp whatsoever, even when I had no pot on the stove and was using it as sort of a "campfire". I'm not sure any stove would perform well in direct rain but even a small roof should suffice, just watch the height of your roof.
That night my starter wood came from a dense grove of pine trees. I snapped off small, low twigs and whittled some shavings into the 5 driest twigs. Still, I doubt the fire would have caught without the dead pine needles collected from low on the same trees. Feeding the fire with the needles 4 times had the fire going strong. Slightly damp wood and pine cones then burned reliably with some smoke.
I don't worry about finding dry enough sources of wood under most any condition here in the midwest where fuel is abundant. I will always carry half of a Strike-a-Fire stick which will get six fires going for under 1/2 oz. I haven't needed them yet. Personally, I am comfortable using the BB UL under a tarp with a high pitch.
Love this little stove. Hope you keep your experiences with the BB coming.