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The Chinese Grammar Wiki writes:
Note that 难道 can only be used for rhetorical questions which could be answered with yes or no, and generally end with 吗.
And gives examples such as:
今天是我的生日，难道你不知道吗？
However, none of the examples on 标准教程HSK4上 page 105 include 吗, such as:
怎么有买了这么多饼干和巧克力？难道你不减肥了？
From the examples I've seen, I'm getting the impression that the 吗 at the end can be included or omitted without changing the meaning.
Question: Can we omit the 吗 at the end of a 难道 rhetorical question?
Yes, definitely you can. However, in some situations, with 吗 it sounds more natural and less rude. And sometimes, with 吗 it might sound less criticizing, such as your first example 今天是我的生日，难道你不知道吗?
"今天是我的生日，难道你不知道吗？" (Today is my birthday, could it be you don't know that?) is a rhetorical question
Omitting "吗" and write "今天是我的生日，难道你不知道？" or "难道你不知道今天是我的生日?" is still perfectly acceptable as a rhetorical question, as long as there's a question mark at the end.
Even without "难道" , "今天是我的生日，你不知道吗？" can still be a rhetorical question if you are expected to know. And you are expected to answer "of course I know"
If you are not expected to know, then it is a real question. I really am asking "is it true that you don't know?" and your answer can be "yes, I know" or "no, I don't"
