Well, it says “…what *he* has to do”, so if “he” refers to “everybody”, then it might make sense to use “he” in the tag question. In reality, it’s obvious the exercise was poorly written, to put it politely. For one thing, it’s impossible to know from the context if “he” really refers to “everybody”. For example: A: Do you all know what Jonathan has to do? B: Yes, everybody knows what he (that is Jonathan) has to do. In the above example, the tag question has nothing to do with the pronoun “he”, since the tag question refers back to “everybody”, not to Jonathan. For another thing, unless there’s a compelling reason to use a gendered pronoun, we usually use singular “they” to refer back to words like “everybody”. In this case, we would write, “Everybody knows what they have to do, don’t they?” Singular “they” has been in use for almost as long as plural “they” and singular “you”. The style manuals of most publications recommend it these days, and most native speakers use it frequently. Here are some articles from Merriam Webster and from the Oxford English Dictionary blog: (I'm not including the links, as italki has been known to punish users for posting links. You can google them, search the Webster and Oxford Dictionaries online, or PM me.)

This is what I know: It's written in present tense, so the question tag has to use present tense. (The main sentence and the question tag must be written in the same tense) Will you / Won't you is used when the main sentence is an exclamation (Ex: Let's go, won't you?) When the main sentence is a positive sentence, the question tag has to be negative, vice versa (This is the grammar rule, I think this rule doesn't always apply when speaking casually)

This tag question is questioning the main clause "everybody knows", therefore, use "doesn't he?" 主句第一人称表观点的时候，反义疑问句问的是从句 其他情况下，反义疑问句问的是主句。