Well, you really have to be an extraordinary applicant to be accepted to a top 20 business school. You must remember that PhD is much more competitive than MBA, for example. Total enrollment for HBS MBA is almost 2,000, while for HBS PhD is only 147 (considering all students of all years of all concentrations, only 24 enrolled every year).
When I see an applicant aiming only at top 20 in business PhD, I think that applicant is probably underestimating the PhD application process. Harvard probably accepts about 2 applicants each year for a specific concentration. Do you really think you are among the 2 best applicants Harvard can get in your field?
And, in your case, you must also see that Health Policy and Management PhD may not be in the business schools of some universities, but other schools with different requirements (GMAT not accepted, for example).
So, let me try to answer your questions:
1) For top 20, I think your GMAT should be at least 730, ideally more than that.
2) What is a top-tier for you? For PhD, research fit is often more important than rankings, for example (actually, there are no good rankings for PhD, and you shouldn't use MBA rankings). Some of the best schools for your specific research interests may be ranked lower than 50th, while lots of top 20 schools do not do the kind of research you want to pursue. Anyway, I see very little research in your profile. You wrote a lot about work experience, and that's usually of very little importance for PhD applications (work experience may actually be held against you).
3) Well, that's too much information to write here...
4) Usually, tuition fees are completely covered by all PhD programs in business. I don't even remember what are the tuition fees of the school which accepted me, since I'll not be paying them anyway. About the stipend, few schools inform the stipend value. And even when they do, there may be changes when you are actually accepted. You can try this site here to have a general idea
https://www.phdstipends.com/resultsI suggest you take a look at my blog, since there I'm better able to answer those questions at length. My concentration is Marketing, but I think most posts will also apply to your case.
https://myphdtales.blogspot.com.br/p/blog-page.html