My dream position would be at that Top Tier University Hospital program, and I am a making a full-court press towards that effort.
1) Would your programs still likely filter me out despite my application?
Yes, sorry, but there are too many other qualified applicants. No matter how you try to spin it, you are a Caribbean grad from a community program. Something happened to you along the way and you fell off the path. Your "top 10" undergrad is largely irrelevant at this point. Your goals of a "top tier" cardiology program just are not realistic. You should be focussing on getting a cards spot somewhere.
2) Besides interest emails, are there other steps I can take to improve my chances of getting an interview?
Not really, you are who you are at this point. Fill out your application thoughtfully, write a good personal statement, and apply broadly.
3) I have mentors who either trained or were affiliated with this University Hospital in the past, and have offered to make phone calls on my behalf. When would it be appropriate and advantageous for them to make these calls? Now, or end of July/early August to give me the best chance of getting my foot in the door for an interview, or in late October prior to the rank list?
The phone call in October will not help you if you don't get an interview in the first place. Also, unless your mentors are friends with the cardiology PD, it will likely end up just being an awkward phone call for your mentors and not get you an interview.
I may be overthinking this, but I want to make sure I play my cards right (no pun intended
). Thank you again for all the advice.