There is a ton of information at The Stanford CS Admissions Page, so please look there before emailing me. (If you want to get a PhD, you will have to be good at research, so you might as well start practicing now! :D)
A bachelor's degree in CS. More detailed information is at The Stanford CS Admissions Page.
Having publications in internationally recognized journals and conferences is a bonus. Have done research previously is nice as well. You will also need strong recommendations.
Write about topics of research that fascinate you, that intrigue you. Your enthusiasm will come through to the admissions committee.
I highly recommend applying for fellowships like the NSF Graduate Research Fellowships. Obviously a student who is "free" is more attractive than one who requires funding.
My parents are from India, but I was brought up in the U.S. So unfortunately, I cannot give you any useful advice about whether your program of study in India is adequate.
I am not familiar with any international exams. As a rule of thumb, you will need to score in at least the top 90% of any test to be considered for admission. I don't know what the current requirements are (See The Stanford CS Admissions Page), but you need to do well on the General GRE, Computer Science GRE, and TOEFL.
If you are a Master's student, you can try to get a TAship, or get an RAship from a willing professor. Note that for Master's candidates either of these are often hard to get, so try to find a professor whose research meshes well with your qualifications. You can find a directory of professors and their interests over at the research page.
On the other hand, the department funds the Ph.D. candidates for the first quarter, after which he is expected to find an advisor, who usually funds them for the rest of their stay at Stanford.
The Master of Science program is separate from the PhD program. While PhD students generally are funded and do not have to worry about tuition, Masters generally do have to pay tuition.
Admission to the MS program does not guarantee eventual admission to the PhD program, although in special cases it can provide a nice stepping stone. For example, if you do great research for a Stanford CS professor as a MS student, then you will increase your chances of getting into the PhD program later on. But this is still no guarantee.
You need to be in at least the top 10% percentile of test scorers.
The research page contains a good, up to date summary on our faculty, and their proficiencies.
Other than Stanford and CSLI, I recommend these institutions: