Is drawing the same character but calling it something different illegal?

It's not illegal to draw anything. You can draw or paint anything you want for your own personal use. Distributing it to other people or particularly, selling it for a profit, is where you can potentially get into legal hot water. Even then it usually has to be reasonably high-profile to get into trouble (people post fan art online all the time and nothing happens, mostly as the copyright holders don't really care).

The other thing to consider is that copyright law itself isn't totally clear-cut on that. Generally, the rule is that individual characters can be copyrighted but ideas and concepts cannot (so for instance, you are free to create a character that flies around in a cape and fights crime, as long as you don't call him Superman). If another character looks very similar, it would basically come down to whether or not the courts felt that it was indeed the same person or character.

Yes, it's illegal if it's a derivative work of something that is still copyrighted. There are nice questions of law and fact that go into such a determination, as not every element of every drawing is copyrightable.

If it's an exact copy, then basically copyright infringement, but kind of tough to prove.

Yes. Anytime you take on a project "inspired" by a mainstream copyright, you'd better hire (and consult with) a good lawyer first.

He needs to make it significantly different. Tell him to improve on the character by making 3 revisions to it. If he is creative, he can improve on what he currently has. Tell him to get out a dictionary, read the thesaurus for the words that describe the character, then find a better word than anxiety and draw it.