Generally, such participial verb forms begin the phrase: "Being an unexpectedly pleasant day..." However, this requires that the participle verb form applies to the subject of the independent clause. In the example sentence, "students" is the subject; and the "students" were not "being an unexpectedly pleasant day." In this case, rewording is often necessary: "Since it was an unexpectedly pleasant day, the students decided to read their books outdoors." Another approach might be to try to choose a participle that matches the subject; but this can be tricky, even for native speakers: "Given the unexpectedly pleasant day, the students decided to read their books outdoors."

"With it being" sounds more natural and also grammatically correct, but I think the use of "expectedly" defeats the meaning of that narrative tool. "It being" has a more matter-of-fact attitude attached to whatever is being described without an expectation, in my opinion. Try out these two beginning clauses to that sentence: - On such an expectedly pleasant day, ... - With it being such a pleasant day, ...

They do mean the same thing, but it being doesn't sound natural. It was an expectedly pleasant day, so the students decided to read their books outdoors. This sounds more natural.

Yes, exactly! "It being..." sounds very literary and not at all conversational.