Relative(s) piggybacking questions

If I understand correctly, Luxembourg introduced a modified Law of nationality in 2015/2016? which allows children of Luxembourgers to acquire the nationality without going through the re-acquisition process (article 29-Law of 2008), If your father is the direct child of a Luxembourger he could apply through the new process. If not you could still apply under the article 29 however adding the condition that he is unable to travel for medical reasons, its possible that they might forgo the necessity to go there for the phase 2, you should give the hotline a call and ask them if its possible.

Would there be any benefit to having him just "on the books" as a descendant or a direct relationship to a citzen (me), if I can't actually get him to go over there for the Bierger Center portion?
No, benefits (or obligations) only apply to the holder of the nationality.

Does me having the citizenship convey anything to my mom, if I end up living in Luxembourg and becoming her caregiver in the future (she's disabled)?
No, at this point if she is not of Luxembourg decent it does not apply to her, nor being the spouse of a Luxembourger.

Is a passport absolutely required for Phase I?
Yes, you need a notarized copy for the Phase I application, as well as to travel.

And, am I going to have a bad time if I've been putting off getting my Luxembourg passport/ID until I'm over there again?
No, you are registered as a citizen, you can apply at any time for the passport/ID card, you don't need to do it at all.

Hope that helps