Property between totally disconnected and zero dimensional
Solution 1:
No two of the three properties are equivalent for Hausdorff spaces in general. In this answer Martin Sleziak gives a proof that the Erdős space $\ell^2\cap\Bbb Q^\omega$, the space of square-summable sequence of rational numbers, has (B) but not (A). The space obtained by removing the dispersion point $p=\left\langle\frac12,\frac12\right\rangle$ from the Knaster-Kuratowski fan has (C) but not (B): its quasicomponents are the segments $L_c\setminus\{p\}$ for $c\in C$, the middle-thirds Cantor set.