Claude, the invisible man, is said to take on the role as Peter's mentor, so I think we can presume that he has greater control over his powers than some of the others we've seen and more control than Peter's got.

Peter was able to see him because Peter's mimickry power works whenever he's in proximity to someone with powers. We can presume that since Peter could see Claude that Claude can see himself when he's invisible.

From what I've seen, at least to start out this half of the season, Peter feels he's the cause of the explosion in NYC and so he wants to find a way to get rid of his powers (so that he won't explode and destroy NY).

We all wear a green carnation.