In general the answer to both questions is no: there are pseudocompact zero-dimensional spaces ([see, e.g., this answer][1]) whose Cech-Stone compactifications are not zero-dimensional. For an $X$ like that one has $C(X)=C_b(X)$ and this ring is isomorphic to $C(\beta X)$.


  [1]: https://mathoverflow.net/questions/93719/0-dimensional-locally-compact-space