As others have noted, we discourage long lists of questions. It's better to post separate questions that are meaty enough to stand on their own. But I'll take a stab at it (disclaimer: not a mathematician).
One of the US university ask only SoP describing background and interests, no CV at all. Should I write all necessary details like my past achievements, which course I have attended(sort intro to the key topics of these syllabi), my CGPA, or percentage of marks in my SoP?
Usually they will request your transcripts, which contain this information. Still, describing past achievements is usually appropriate, as this describes the foundation for your future plans.
Should I include any professors(of the university where I am going to apply) name and his work(some articles and books) that I used in my past study?
There are a lot of posts on this site about how and when to mention specific professors that you might hope to work with. Being familiar with a professor's textbook or article might be worth discussing with that professor, if you reach out to them. But it's a bit strange to mention it in an SoP, unless you work it in as a "fun fact" rather than a serious topic.
Is there any rule of decorating the SoP, I mean use of margin, font size, space between paragraphs. I asked this question because, in some lines, I have to write some mathematical notation(contained in the name of a topic), and in this case, I have to use Latex. So is there any problem If I submit it as a .pdf file made by latex language?
This is entirely down to your university's rules. I would expect that most would accept a PDF. But I would also suggest that mathematical notation should be kept to a minimum (and Microsoft Word is able to handle fairly sophisticated mathematical notation, though it is painful).
Should I include any sentence like, "I have plan to take some courses on..... in the upcoming semester''? Here upcoming means, if a grad school selects me, then the gap between the joining and current semester.
It can't hurt, but again, this should not be the focus of your SoP. Your SoP should focus on your achievements, interests, and plans. I suggests reading some other posts about SoPs on this site -- there are lots of common traps (e.g., far too many SoPs talk about inane childhood stories; this is a professional document and should focus on your professional interests and plans).
If some grad school offers an independent study program, should I write a sentence like " the independent study program of your university is also a flexible way of enhancing my knowledge required for research"? I know that I am intended to join as a research student, but still, not all requirements can not be completed just after passing the MS degree, at least in case myself.
I don't really understand why a PhD program would offer independent study. In any case, mentioning your familiarity and approval of any university's particularities is a good thing. But I would work on the writing -- your sentence is very long and contains no real information other than your familiarity with this requirement's existence.
After completing the MS program, I have joined some university in the home country as a regular Ph.D. student. So there is a one year gap between passing MS degree and applying for US university. So should I write what I have done in this gap year of that particular university? I have done some credit courses (in this gap year) that are helpful for research in my domain.
Sure, this is part of your narrative. But I would expect them to require a transcript from this school, so you don't need to dwell on it.
Finally, one unsolicited comment: I would seriously consider hiring an editing service (ideally one familiar with academic norms) to help you revise your document. In my experience (and basing this off the writing in your question), a good editor would be able to take 4-5 pages from you and help you cut it down to a very well-written 1.5-2 pages, just by deleting and rewording. And you will learn a lot from watching them do this; this is a very valuable skill.