People here on this web site can give you some ideas and suggest some specific universities for you to consider.
If you attend university in the US, then you should assume that you will need to return to your home country after graduation. This leads to the question…
…Are there jobs in Mongolia for someone with a degree in computer science from a university in the US?
Another issue is that there are a lot of universities that are very good for computer science, and in some cases these are not necessarily the same as the universities that are the highest ranked overall.
There is some overlap. Stanford for example is one of the highest ranked universities overall, and is also one of the highest ranked universities for computer science (and for mathematics, which is a talent and interest that often co-exists with CS in the same student). Stanford also has very good need based financial aid for students who qualify. However, it does consider need when deciding whether to accept any particular international student, and you would need to be very close to the top high school student in your country to have a realistic chance to get accepted as an international student at Stanford.
However, the University of Massachusetts in Amherst is also very good for CS, as is Rutgers and quite a few other universities. These would be easier to get accepted to compared to NYU, Northwestern, or Stanford.
And you do not need to apply ED in order to get accepted to very good universities in the US. I am not a fan of ED unless both of the following are true: (1) a university is your clear top choice; (2) you know for sure that you can afford to attend if you are accepted. If either of these are not true, then you can apply either EA or RD to multiple schools and just wait and see what sort of offers you get. It does not seem to me that you have a clear top choice at this point.
There are also some universities in Canada that are very good for computer science. Waterloo may be the best known for CS, but Toronto, UBC, Alberta, and McGill are also very good (and coming from Mongolia you might even be ready for Alberta winters!). Simon Fraser University (in British Columbia) is also quite good for CS, as are U.Victoria and U.Calgary. Admissions at these schools is more likely. Financial aid is unlikely at schools in Canada, but the cost of attendance is typically lower compared to the near US$100,000 cost of some universities in the USA. Being able to stay in Canada after graduation from university in Canada is unclear, but is probably more likely compared to being able to stay in the US after graduating from a school in the US.