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.
The second question appears to be more involved than the existing answer indicates.
It is possible that your school may own the copyright (at least) to your work.
See MIT Policies and Procedures | 13.1 Ownership of Intellectual Property:
Copyright ownership of theses generated by research that is performed in whole or in part by the student with financial support in the form of wages, salaries, stipend, or grant from funds administered by the Institute shall be determined in accordance with the terms of the support agreement, or in the absence of such terms, shall become the property of the Institute.
It is further possible that your funding source may have a larger stake in your intellectual property.
See Do I Own My Dissertation? — Columbia Copyright Advisory Office:
For instance, an employer supporting the student’s studies or an outside funding source may lay claim to whatever intellectual property is created.
Although the above source notes that this is not the usual case, a graduate student acquaintance of mine was not even allowed to continue work toward his thesis project after sale of his research by the university to an outside corporation. He had to choose a new project and begin again from scratch.
