Does your university have a honors college? My honors college has all sorts of information
Fulbright posts their statistics online, and they are far, far less competitive than Rhodes, Gates, Marshall, etc., with the exception of a few countries (UK, western european countries). For example, by the numbers, the UK open award is actually more competitive than Rhodes (2/165=1.2% for UK open award vs 32/2500=1.3% for US Rhodes). However, most Fulbright acceptance rates are closer to 20-40%. The competitive countries are a bit less than this (5-20%; e.g UK is 5% overall, Japan 30%, France 15%). They are still very impressive and well known awards, and would be a great way to get involved in an underserved country (hint hint). They do provide a funded way to study (research grant) or teach English (ETA) virtually anywhere in the world, but are more about cultural exchange, so you should really choose this path if you have a specific reason for going there. To be successful they require a strong connection to the country, so you probably need to get ahold of a professor ASAP and start writing a statement. I was selected as an alternate awardee ('first runner up') for the UK open award, so I could answer more about this if you'd like.
https://us.fulbrightonline.org/study-research-eta-statistics
Rhodes is the most prestigious award there is at this level (to Oxford). The most notable winner is probably Bill Clinton. The Wikipedia page is fairly helpful in putting it into context.
Rhodes Scholarship - Wikipedia
Marshall is extremely competitive and places a lot of emphasis on sheer numbers and "ambassadorial potential". They require 2 schools, including 1 non-elite (not Oxford, UCL, Cambridge, Kings, Imperial).
Churchill is to Cambridge and should also be on your differential. They tend to select from nosebleed schools (i.e., T20's).
Gates is more service oriented and is about how you'll help humanity. This lends itself really well to pre-med applicants. Average GPA 3.93.
Hm, I was really more ambitious when I began writing this but I need to get to lab. Check your college page. I'll update this if there is sufficient interest.