How can I recover professionally from being a crank?
A few suggestions:

If it were me, I would be very forthcoming about the illness. Of course, no one is entitled to your private medical information, so you will have to decide what you're comfortable with. But &quot;an undiagnosed brain tumor caused me to act erratically&quot; is a very convincing explanation, and does not reflect poorly on you.
Consider reaching out to your previous advisors / colleagues. Even those who were furious at your previous behavior would probably accept that an undiagnosed brain tumor caused you to exhibit poor judgment. It's a &quot;Dr. Jekyll and Mr. Hyde&quot; situation.
Along similar lines, consider contacting the university, and the specific school within the university, to discuss whether the expulsion can be expunged or converted to some other classification. Most universities are really understanding if you speak to the right people.
When you apply, I would absolutely explain what happened. I would give a shorter version of what you wrote above -- even if they don't recognize your name, they will see that you were expelled, so it's better to tell your story. And you did a good job telling the story above, so I trust that you can write an explanation that is similarly clear and compelling.
There will likely be no need to provide medical proof. You could (perhaps even should) offer to provide proof, but I suspect most admissions committees would not request it. Among other reasons, it will likely be clear from your tone that you have recovered.
I don't think it's necessary to send an apology to your entire mailing list. But I don't think doing so would be inappropriate either. If you do so, be concise: no need for more than a sentence or two.