Can a footnote be an incomplete sentence? For example, a footnote for "Pharaoh" says, "The Egyptian king during the time of Prophet Moses."
If so, should "the" be capitalized or not?
Any annotation whether footnotes or elsewhere allows any information at all inside conceptually. What you want to consider is if you are providing context.
Consider a dictionary entry, which use a similar declarative style to your provided example. Another common example would be Wikipedia citations.
The Egyptian king during the time of Prophet Moses
If this footnote is numbered and matches a superscript number in the document, then you have created a fairly standard "reference". This provides a definition of whatever you're noting.
There's nothing grammatically "wrong" with this, you already understand this is not a complete sentence. So you cannot use it as a sentence which, by-and-large, is what grammar is for. A note does not need to be a sentence in a document.
What you'll instead need to check for is a style guide provided for the type of document you are submitting. If the document is for your own use, business letter, general publishing, etc.. then you are free to use your own style guide.
The starting letter is conventionally capitalized. One could make an argument this is not technically required. There are instances in programming texts for instance where note blocks begin with
I am in the process of writing a lengthy family history - my children refer to it as a history of the western world with passing references to our ancestors - and, rather than just make bald assertions of what happened where and when, I give the source for pretty well every sentence. The work would be absolutely unreadable if that material was incorporated into the text. In the result, footnoted comments are complete sentences ("Read takes a different view: see page 27 of his work.") but the vast majority of my footnotes are not comments and so are not complete sentences ("Upper Canada Sundries, page 58753, letter from Baker to the Lieutenant Governor"). The same is true of the footnotes in pretty well every commercially-published history book that I have seen, and I have seen many. My only question is whether I should end mere sentence fragments with a period; logic says "no" but it looks strange not to do so.