Learn Farsi: در مورد آن شایعه چه می‌دانی؟

It translates to "about" or "regarding." It literally means "in the subject," so to connect it to the next phrase (ân shâye'e), it needs an invisible ezâfe (an "e" sound) at the end of mored. You pronounce the whole phrase as dar mored-e ân shâye'e.

Can I use râje' be instead of dar mored-e in this sentence?

Yes! Both mean "about." You could say râje' be ân shâye'e che mi-dâni? The only difference in grammar is that dar mored connects to the next word with an ezâfe (an "e" sound), while râje' be uses the preposition be (to) instead of an ezâfe.

Why doesn't the word ân get an ezâfe to connect it to shâye'e?

Demonstrative words like ân (that) and in (this) do not take an ezâfe. When they come before a noun, they sit right next to it without needing that extra linking sound.

Is mi-dâni present continuous ("are you knowing") or simple present ("do you know")?

It translates best as simple present ("do you know"). In Farsi, the prefix mi- is used for both simple present (I know) and present continuous (I am knowing). Since "knowing" isn't an action you usually do continuously, it naturally means "do you know" here.