1. For CuSO$_4\cdot5$H$_2$O: Four H$_2$O molecules coordinate with Cu$^{2+}$ in a square planar arrangement, and one is hydrogen-bonded to SO$_4^{2-}$. Thus, statement I is incorrect as more than one H$_2$O is involved in hydrogen bonding. 
2. For BaCl$_2\cdot2$H$_2$O: Water molecules occupy interstitial positions in the lattice, not coordinated to Ba$^{2+}$, making statement II correct. 
3. For CrCl$_3\cdot6$H$_2$O: Typically [Cr(H$_2$O)$_4$Cl$_2$]Cl$\cdot2$H$_2$O, where four H$_2$O molecules coordinate with Cr$^{3+}$. However, in some forms like [Cr(H$_2$O)$_3$Cl$_3$]$\cdot3$H$_2$O, three H$_2$O coordinate, making statement III correct in context. 
4. Thus, statements II and III are correct, so the answer is (3) II, III only.