Participial constructions using "it"

You can compare sentence 2 with its full-blown analogue:

a. It being hot, I opened the window.

b. It was hot, so I opened the window.

Obviously, sentence b contains two key information which are lacking in the first one: "was" indicates the past tense, and "so" indicates the meaning relation between the two clauses (the hot temperature is the reason for my opening the window). Both these information are guessable in sentence 1, but they are not clearly stated. For this reason, sentence b is more easily understandable. Also, with the sentence given outside the context, some confusion may arise in interpreting the subject "it" as either a dummy or a pronoun referring to something in the previous context. (the latter interpretation being unlikely with the main clause added)

That said, I don't mind "It being hot, I opened the window", if style considerations justify chosing it over the clearer: "It was hot..so". However, even in that case, I'd probably go for the more precise: The room being hot, I opened the window.

Your second sentence seems perfectly natural to me. It's an absolute phrase construction, where the first part gives the reason for the second part. We sometimes use the pronoun "it" to refer to the weather, temperature, time, duration, distance and the like. So, your second sentence basically says: since the room was very hot or the room temperature was very high, I opened the window. As for "It being the holiday season, the train was full.", the "it" here represents time. The first part explains the second part, just like in the previous example.

In "Feeling hot, I opened the window" and "It being hot, I opened the window" ", "Feeling hot" and "It being hot" are free modifiers. These act adverbially on the whole clause "I opened the window." See http://writingwithaesop.blogspot.com/2011/01/commas-and-free-modifiers.html