Ceiling – Could there be asbestos in the thick brown paper / card backing above the popcorn ceiling

Vermiculite on its own poses no health risk, and can be removed without concern. However, in many cases, mined vermiculite can contain traces of Asbestos, and so its removal is not recommended without an asbestos test. I would, as Scott suggests, call the testing company, and make sure that since no asbestos fibers were found, this indicates that the vermiculite used was asbestos free. If that is the case, you are good to go (that being said- wear a mask when you strip the popcorn ceilings, or you'll be coughing and hacking a lot- that dust is nasty).

My thoughts on your first question would be to take a sample from a few different rooms, if you can see any differences between color/texture on the ceilings. From asbestos.com:

If the material looks different (especially in color or texture) in different parts of the room, plan to sample each of those areas. For a ceiling surface in a large room, two samples may be needed to determine your asbestos situation. Try to balance the need to protect your family from exposure after you sample (to sample in an inconspicuous area) with the need to get a representative sample of the material (to sample in the center of the room).

I'm not sure about the size of sample required, but I don't think you'd need more than a few square inches.

As for patching the ceiling, you should prime the sample spot before painting. After that, you can mix some standard ceiling paint with a texture additive to match the consistency of your original ceiling. Since it's a small area, you can probably get away with using a brush, or possibly a small roller. Or, instead of bothering with texture matching, you could go the easy route and use spray-on popcorn ceiling texture after priming (beware the horrible product reviews though - it might be easier to use a pre-mixed product).