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.