We can't "help you with all of these questions"; that would require a full course in relativity, and that's way beyond PF's ability to provide.Doesn't change my initial question to please help me with all of these questions ?
You have already received a number of responses that indicate, first, that at least one item in the full question (item e) doesn't make sense as stated, and second, that you appear to lack important background knowledge that is crucial for understanding the question you're trying to answer (let alone actually answering it). Those are not problems that can be solved by you continuing to ask questions here. You need to provide more information, such as what I've already asked: what textbook is this problem from, and what chapter/section/problem is it? That might at least help with the first issue (that item e in the question doesn't make sense as stated). The only way to help with the second issue (your background knowledge) is for you to put in the work to learn the basic concepts involved in the question (such as what I suggested, Killing vectors), which, as I said, is beyond the scope of what PF can provide.