Let $\mathcal{C}$ be a small category and $X,Y$ objects within.
If $X$ and $Y$ (as $0$-cells) lie within the same path-component in $B\mathcal{C}$, can one say anything in general on how $X$ and $Y$ are related in $\mathcal{C}$?
This is a follow up to this question. I had thought that maybe there might exist a morphism from one of the objects to the other, but this is false. Is it maybe true that there exists a sequence of objects $X = Z_0, Z_1, \dots, Z_n = Y$ such that there exists an arrow $Z_i \to Z_{i+1}$ or $Z_i \leftarrow Z_{i+1}$ for each $i$?
What you describe in the last paragraph is the equivalence relation of being connected on the class of objects of a category. Every category is the disjoint union of its connected components, and the classifying space preserves disjoint unions, as can be checked directly. The answer to your question is therefore: Yes.
This argument also shows (as confirmed by the nlab article on connected categories) that a category is connected if and only if its classifying space is connected.