Suppose I have a genus 1 curve $C$ over a field $k$. If $C$ has a point, then we can embed it into the projective plane by a Weierstrass equation. Now let us suppose that $C$ does not have a point (so that it is a non trivial torsor for it's Picard group).

Can I still embed $C$ into the projective plane? I guess not but there is apparently a theorem of Lang-Tate that we can always find an effective divisor of some degree over $k$ (what is a reference?) so we can embed it into some high dimensional projective space.

Can we always embed C into a Severi Brauer variety of dimension $2$?

embedsinto the projective plane then it does so as a cubic. Hence the answer to your question is no unless the curve has a divisor of degree $3$ (in which case the answer is yes, using sections of that divisor). $\endgroup$