Is the vermiculite found in the popcorn ceiling toxic? Can I remove it theself

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).

Check out this PDF from the Australian government (assuming the +61 in your profile means OZ)

Pages 29 and 30 deal with personal protective equipment. (PDF pages 42-43 AND WHY CAN"T ANY MAKE PDF page numbers work!!!)

Asbestos Removal Guidlines

I'll highlight this line: The laundering of contaminated protective clothing in workers’ homes is strictly prohibited.

Generally though, this is a workplace safety issue, and I'd contact the Ministry of Labour.