I'm currently applying for observational astro PhD programs, and I can tell you what I've heard from professors, admitted grad students, etc. Schools look for different things in an applicant, but there are some general rules I think are usually true. Yes, you have to take the physics GRE, but your score doesn't need to be nearly as high as for physics programs. With a good GPA and good research experience, no red flags will go up if you score 50-60th percentile and above. Of course, that may not be true for the most competitive schools. I've heard that Caltech astronomy won't even look at your application if you get below a 70th percentile.
About funding, I'm not sure if whether what I know applies to international students. For American students, the standard for astro grad school is: tuition fully covered, and around a $30K stipend a year.