Is it permitted to cite works by members of my own family, in classwork? I'm starting a unit (undergraduate class) at my university which deals very much with topics researched by my grandfather - He has written a number of books and papers on the issue.

I don't yet know if I'll need to cite them but I thought it would be wise to ask now, rather than with a looming deadline. Is there anything I should take into consideration if I end up citing his works?

Additionally, would it be considered unethical (and/or an 'unfair advantage') to clarify anything on the subject matter with him, or does that just constitute "good research"? If I did get clarification from him, would I have to add my communications with him in an appendix?
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.