The answer to the first pair of questions is that there's no reason that such cycles should be homeomorphic. The question about algebraic subvarieties also has a negative answer, but presumably more restrictive questions might have more positive answers.
Let's stick to the case when the ambient space $X$ is an $n$-manifold, and let $d < n$ be the dimension for your homology class. Then you can simply take $Z$ to be the union of $Y$ with a $d$-dimensional sphere lying in a ball disjoint from $Y$. The number of components has changed so $Y$ and $Z$ are not homeomorphic; on the other hand the sphere is null-homologous so you haven't changed the homology class.
The other question depends on context. If you really mean subvariety, then you presumably allow some singularities; there are many examples where a non-singular curve can degenerate into a singular curve; this doesn't change the homology class. A simple example would be in $CP^2$, where a smooth curve of degree 2 (thus homeomorphic to a 2-sphere) can degenerate into a union of two lines (two 2-spheres meeting at a point). If you intended that your subvarieties be smooth, then this is more subtle and perhaps someone more knowledgeable about Chow rings can answer your question.