Why does a molecule with a Thiol group form Hydrogen bonds?
Actually it's all four.  See here for I.  The article describes a hydrogen bonding interaction between methane and water.  It's weaker than the "usual" hydrogen bonds but still contributes to forming methane-water clathrates.

The larger lesson here is that hydrogen bonding is not just the electrostatic interaction often described in textbooks.  It has a molecular orbital component that works even with primary bonds of low polarity.