Proving whether $\bf{K}$(Happy $\lor$Sad) $\implies$ $\neg \bf{K}($Happy) is satisfiable, valid or unsatisfiable
The formula is either satisfiable or unsatisfiable. Only when it is satisfiable, it may also be valid.

To rule out that the implication is valid, show that its negation is satisfiable, that is, construct a model that satisfies the LHS but not the RHS. (Hint: a model with just one possible world suffices.) 

Next, rule out that the implication is unsatisfiable by providing a model that satisfies the RHS. (Hint: again, a model with just one possible world will do.) 

After these two steps, you're left with the answer.