If I have two topological spaces $X$ and $Y$, and I stipulate that $X$ can be embedded in $Y$ and likewise $Y$ can be embedded in $X$, is that equivalent to saying that there is a homeomorphism between the two?
If this does not make sense, please explain where you think I have gone wrong in my understanding of what an embedding is/means.
If the question makes sense, but its claim is not true,
1) Is there some set of properties that $X$ and $Y$ could have which would make it true?
2) Is there a well-defined minimal set of properties needed for a 'half-homeomorphism', i.e. if you have these in one direction and also have them in the other then a homeomorphism exists.
Define $X=\mathbb{N}\times( S^1\times S^1)$ and $Y=(0,1)\sqcup X$. Now fix an embedding $f\colon (0,1) \to S^1\times S^1$ . Then it is easy to see that $X$ embedds into $Y$ and that $Y$ embedds into $X$ as follows: an element $x$ in the $(0,1)$-component gets sent to $(f(x),0)$ and an element $(x,t)$ in the $X$ part gets sent to $(x,t+1)$.
Nevertheless it is quite easy to show that these two spaces are not homeomorphic as every connected component of $X$ is compact, while this is not true for $Y$.
Regarding your other questions I'm not so sure if nice criteria exists for that. Maybe compact is enough but I'm still skeptical.