What kind of help can I request from a terminally ill professor?
Edit: note that my original answer below addresses OP’s original version of the question, which has now been edited. 

To address the current version of the question: yes, it is okay to ask for help, but your motivation should be wanting to get help, not artificially creating feelings of “being appreciated”. If you want to let someone know they are appreciated, just tell them that that’s the case. And if you don’t know what questions you should ask, to me that is a sign that there isn’t really anything that you particularly need help with, so the best thing to do is to not ask any questions.

Finally, the fact that you know the professor is ill shouldn’t change anything about the nature of the questions you should ask him. Just ask what you want to know, whether it’s an easy or difficult question to answer. I am confident that if the professor finds the question too burdensome to answer given his illness, he will either not respond or will simply tell you that he is unable to help.



My original answer:

Maybe I’m misunderstanding your intent, but the way your question is phrased (particularly your statement “my main goal here is to make the professor feel appreciated” and the slightly absurd list of hypothetical questions you are thinking of asking the professor) I get the impression that you’re thinking of inventing a fake request for help, that is, a question whose answer doesn’t actually interest you very much, out of the belief that the professor will feel “appreciated” when receiving the question.

If this interpretation is correct, my strong advice would be: no, it’s not okay, so don’t do it. The reason is that this would be dishonest (and somewhat transparently so, for some of the questions you proposed - not to mention that you’re publicly posting your plan here using your real name...). I can’t claim to know what it feels like to be terminally ill, but as a general rule people don’t appreciate other people being dishonest with them, so I think your question could end up having quite a different effect than what you intend.

Here’s an alternative suggestion however: if you want the professor to feel appreciated, why not simply tell him that you appreciate him (assuming that’s correct)? And certainly I think it’s okay to offer to help him in any way you can — again, assuming the offer is genuine and you intend to follow up in case he takes you up on it. It’s also okay to ask him a question if you genuinely want to know the answer, but you should be asking it because you want to know the answer. Please don’t invent questions solely for the purpose of making someone feel appreciated; I find that idea patronizing, offensive, and highly misguided, despite the fact that the intentions behind it are good.