Yes, the word "hurt" can be a noun; here's, as I think, a good example of its usage as a noun that will make it clear: "She saw the hurt in his eyes."

In the example you have given, the word should be 'pain'. The pain in my arm seems to be increasing/ getting worse. When you are talking about physical pain, 'hurt' cannot be a noun. However, we do sometimes use the word 'hurt' as a noun when we are talking about emotional pain or damage, but this is more unusual.

Yes, it can, when referring to it as a thing. For example: The hurt she caused him. In this case, 'hurt' would be a noun.

This is understandable however for a smoother sentence use pain so "The pain in my arm seems to be growing bigger" Hurt can be a verb or a noun. For example as a verb: I hurt him or they hurt the dog As a noun. "The cat is hurt" or "the hurt from the cats pain was horrible" the last one means that I felt hurt that the cat was in pain. Hope this answers your question, all the best