Can a black hole be characterized by an observer who is inside the black hole?

Can any of these parameters be determined by an observer who is inside the event horizon of the black hole?

Yes. As long as they can make non-local observations (i.e. they are not confined to a small windowless spaceship) then an observer inside the event horizon can in principle gather just as much information about the black hole as an external observer. If anything, they might have more information than an external observer because they may have a less distorted/red-shifted view of other objects that have passed the event horizon.