In this sentence, yes, they are interchangeable. Your sentence is perfect, just change humour to humor.

No, they don't have the same meaning at all. In fact, I can't think of any situation in which these phrases would be interchangeable. As Jamie says, 'pick up on' means notice or perceive. 'Get the hang of' means learn to do or to manage to understand something over time. And 'humour' is the correct British spelling.

Not quite. Get a hang of, or get a grasp of means to manage in a certain field, a language for example. Pick up on, here is interchangeable with 'notice'.