If I publish a Master's thesis, does that count as a "publication" (ie protect me from other people publishing my work as their own?)
If I think I can get a paper out of my thesis, am I forfeiting this possibility by sending copies of my thesis around to other people to look at?
In order, yes, and no.
A thesis is a public document and thus helps establish "prior art" in terms of research. Of course sometimes researchers come up with the same ideas simultaneously, or certain work is just not known as widely as it should be, so sometimes work gets innocently repeated (as compared to plagiarised, which can also happen, but reasonably rarely).
As for publication, a thesis is an entirely different kind of publication to a paper, in every discipline I am familiar with, publishing papers drawn from work in a thesis is expected, sometimes even required. The only (?) way you could interfere with further publication would be if you published your thesis as a monograph.