Is it permitted to cite works by members of my own family, in classwork?

I'm not aware of any reason you couldn't cite work of a family member. Indeed, a tenet of academic work is to cite your sources and give references appropriately, so due credit is given to previous works.

Were you not to cite your grandfather (or indeed anyone else's work you reference), that in itself would be the improper behaviour.

It is normal (and often necessary) to cite your own work as well - in order to avoid self-plagiarism, you need to reference any previous works by yourself if you use them in a future work, so as not to be re-presenting the same work repeatedly and claiming it as new every time.

Regarding clarifying points, that wouldn't be a problem - it is actively encouraged to discuss and collaborate with others in research. Authors place their email addresses and affiliations on publications to facilitate private conversation and discussion with other researchers, and often this is how collaborations and future advances happen. Conferences are also organised with the principle aim of facilitating dialogue and further discussion between researchers.

You would be able to cite information from clarifications with him as personal correspondence type references, and I don't see a reason you would need an appendix to quote the actual correspondence.

In the event that there were to be any kind of potential for conflict of interest, you obviously should disclose it as such. Also, if you are significantly referencing personal correspondence with your grandfather, he may be making sufficient intellectual contribution to be considered an author of the paper. Simple clarification of points in already-published works wouldn't be an issue (otherwise I'd be an author on any paper as a result of an email I replied to!), but significant intellectual contribution or addition of new material would probably require you to regard him as an author. For assistance and clarifications which don't constitute author him, you could add an acknowledgement to the paper to express recognition of the assistance and advice given.

Disclaimer - there may be discipline specific nuances here I am not aware of, most probably around how to cite personal correspondence, since it's not something I've had to do before. The requirements to declare authorship may also vary between disciplines and publications based on their regulations.

By all means, do not treat your grandfather (or his works) any different from any other publications you come across when it comes to citing. If the citation is appropriate, do cite them, and if it is not, do not cite them.

There is nothing unfair or unethical about using the opportunity to clarify anything on the subject matter with him. Indeed, I would argue it constitutes "good research" to use such a source of knowledge; there is no point in cutting yourself off of possibly significant information and restricting yourself to what is literally written in a paper just because other people may not have the option to casually call your grandfather to inquire in the case of inclarities. The only thing you may want to make sure is that if a paper you want to cite says X, and your grandfather insists he actually meant Y, be aware that your readers will not be able to find the additional information, so the paper in question may not be the best source to cite Y from.

Lastly, there is no need to add any such communications to the appendix. During your research career, you will come in touch with plenty of people who work in the same field as you do, and you will cite numerous of them. You need to make sure you provide all information you get, and you properly reference where you got it and where other people can check it. However, that is for informative purposes towards your readers, there is no ethical obligation to publicize the specifics of when you talked to whom while conducting your research. Thus, you may have received a particular bit of information in the conversation with your grandfather, but (if you ask him!) he might point you to a work that contains the same bit of information that you can actually verifiably cite.

I would say it's definitely permitted. Citing is a part of preventing bias...the point here is "this comes from a published work, so it's trusted." This is a major reason why you are required to self-cite, too. Citing shows that someone already approved the results that you are referencing in your paper, so people it's not just a quote pulled out of thin air from your crazy uncle.

If you contact him personally and use the information he gives you in your paper, that requires a separate citation. Most reference styles have a format for a "personal interview" or "personal communication."