What are they looking for?/What is the most important part of the application?
I think the questions by Twinnering and evilbooya (the first one) can be equated - they're looking for the thing that is most important about you, which you've hopefully not hidden. The beauty of these kinds of applications is their open-endednes; sure, they're sterilized of emotion and listed as the "personal essay, explanations of past research, project proposal," but it's really just an autobiography, and everyone loves talking about themselves. The application doesn't have any important foci - it has specific boundaries on where you can say certain things. Rather, the important foci are what you lay down, what's important about you. I try to convey a few levels of my personality: my basic curiosity of how things are computed (short anecdotes in introductions are a good place), or my work ethic. I think above all, they are hiring a person, so it's most important that they see that. They can objectively tell how your science is going to be, they have a ton of experts on that, but what they need help understanding, and what you have to guide them to, is that you are a unique student-researcher who will change the world at some point so they might as well start helping you now.
Are the LoRs as important as the true application?
The LoRs corroborate the foundation of your story, which is your past work. No corroboration, no foundation, no true application. This is an endorsement from their peer(s), so there are at least 2 factors (imo) that influence the weight of a LoR: the level of regard of the letter writer and their knowledge of your impact in the project. Best case scenario is a Nobel laureate that befriended you and made you first author on a Nature paper, and the worse case scenario is fresh lab head that only remembers the time you spilled their coffee. Most of us fall in between, so with the quality of the foundation in mind the rest of the application can come together, focusing on the weak spots.
Does taking 2 years off look bad?
Honest answer: I don't know, I can't say because I haven't returned yet and am far from residency applications. I am anticipating that if the years are filled with a vigorous productivity and something tangible like a paper, poster, or presentation then it'll look fine. I also hope to work at a location that fosters this kind of training, so I suppose you can sample bias your choices toward the programs that look for these kinds of applicants. If 2 years is somehow a red flag, then perhaps it'll be enough to prompt them to look closer at your application, so in this view it might serve as a way to stand out.
Which program did you do? What was the experience like?
I'm not done yet so I'd like not to reveal exactly where I am. I'll try to get back to this.
Any recommendations to applicants?
Pick a PI that will offer you an interesting (to you!) project, mentorship (if it's missing you'll hate it because you're lost more than you'd like to admit...), and do the background reading necessary to make a convincing grant. Pretend you are submitting to the NIH as an RO1. Get a past example of a successful one if you can. More than you, the PI, and a few friends should read it - send it to a former PI if you can, or a post-doc or grad student you respect. The point is, get more eyes on it, eyes that are not in that field. They should be able to understand it. If they can't, then the administrators of whichever program you're applying to most likely will not understand it (or have the patience to try). Lots of pictures helps!
What criteria do the HHMI people evaluate the submitted applications?
What everyone who has a **** ton of money does when they want to invest it in a project: make sure there is a strong likelihood of return on investment. This goes for NIH and Doris Duke applications, too. I'm really not trying to be a smart-ass here, but what criteria would you use? These are institutions with money, so it's not a bunch of professors around a table, these are scientific investors, so they want more than an outline of the questions and experiments that can lead to answers. Let's assume the science is sound. What they want is a picture of how thoroughly you've anticipated technique drawbacks, error rates, estimates of the time required for the necessary experiments you've laid out, time required to learn new stuff, time to redo stuff, time to redo it again. What if your project cannot go to completion in the time allotted, can you still deliver something?
If you do not have an HHMI professor as your PI does that affect your application greatly even if you are applying with a strong project proposal with a professor who is well funded by the NIH?
I cannot say from my personal experience, but from what I have heard it does not negatively affect your chances.
In your opinion what makes an application stand out?
2 things: voice and a moderate degree of disinhibition. These reviewers are reading hundreds of applications (where tens of applicants are often from the same school) so it's bound to get somewhat monotonous. A strong voice captures their attention, almost engages them as if there were a speaker present, and I think this can break the monotony and shift a greater amount of their attention and memory to the current application. By disinhibition I mean allowing yourself to geek out, philosophize and think "what if." I think by posing new ways of looking at things you can get the reader to question you, and the point isn't that they agree but that they've stopped and thought about you. And they didn't do this with the previous applications.
Some important things y'all should note and rejoice in, for there is hope for all. I personally had an average MCAT score (f*ckin' standardized tests), middle of the road grades, and no prior publications (though I hope to double that number!..). See, anyone can do it. Good luck in all of your endeavors. Peace on this Earth.