Can I always use "show" to say what people are feeling or are there exceptions?
For example, can I say:
He has shown no happiness?
OR
He has shown that he isn't happy?
We might use "show" that way, but on its own and without context, the first sentence is not idiomatic.
We might say, He has shown he isn't happy by [growling/attempting suicide/...]
but
He has shown no happiness is the sort of thing Daleks might say, inspecting their human captive. It doesn't sound quite human.
It is more human to say, He seems happy/unhappy, or He's clearly/evidently happy/unhappy, or He has given no indication that he is [un]happy, or He has indicated that he is [un]happy.