Below $\phi$ stands for a modal sentence.
The question is to decide whether it is true that 1) if $\phi$ is satisfiable, then $\square \phi$ and $\diamond \phi$ are satisfiable, and 2) if $\phi$ is satisfiable on a transitive model, then the same is true of $\square \phi$.
For 1): I think both are true. Suppose $\phi$ is satisfiable in some world $x$ of some model $M$. Consider the model $M'$ obtained from $M$ by adding an extra world $y$ and the relation $y\to x$. Then $y\models \square\phi\land \diamond \phi$. Is this a correct argument?
Could I prove these claims using the fact that consistent=satisfiable? For example, I tried to prove that if $\diamond\phi$ is not consistent, then $\phi$ is not consistent. The former tells us that $\square \neg \phi$ is provable in K. But I don't see a way to deduce that $\neg \phi$ is provable unless we are working in KT, which is not the case.
For 2): The above argument doesn't work because $M'$ can fail to be transitive. But I couldn't come up with a counterexample.