TOEFL

Hi,
Check University of Texas at Dallas, John Hopkins University, Indiana University (Kelly), University of Southern California, Boston University, University of Washington-Seattle Campus, Missouri University of Science and Technology, Drexel University and more. On the more competitive side you can check Carnegie Mellon, Georgia Tech, and Virginia Tech etc.

Well, it is depending upon that towards which field or work you want your future to be. It also depends on your eligibility to apply for the university. It can even be a College affiliated with a University.

Hello Anuplab, USA most of the universities require GRE and TOEFL scores. There are few universities who don't require GRE scores, but they are not ranked as such. Whereas in UK, you will have to only go for IELTS exam. GRE is not required for MS. Also USA gives you OPT but UK do not have post study work visa.

Dear Ishan,
As per their admission policy on website, if the duration of your Bachelors is less than 4 years then you are not eligible to do MBA from Harvard or Stanford because they generally follow American education system. According to that, the length of graduation is 4 years (fresher, sophomore, junior, senior). Hence they wouldn't allow if you have done B.Com, BBA, BA or something like this.
In addition to this, according to them if you really wish to pursue MBA from Harvard, you need to have post graduate Diploma after Bachelors. IELTS or TOEFL is only required if the medium of instruction in your college is not in English.

Hi,
Yes, either TOEFL or IELTS. If the programme delivery language is German you must have German language fluency too.

TOEFL tests your English conversation skills. It is absolutely necessary because the university wants to know whether you will be able to understand what you are being taught.

Without these tests, you will not get admission in a good University. These tests are necessary to understand your reasoning and English language skills.

Yes, these exams are for higher studies in foreign countries like MS, MBA, etc.