iwtlhf

To answer the original question - why you have not received personal answers regarding the dismissal of your application, there are a multitude of possibilities. Foremost, professors may have simply decided to ignore the email. No one will be able to explain why this is the case, but it is certainly a possibility. It also seems that the correspondence with potential advisors was rather impersonal. From what you have suggested, the professors only said they would consider mentoring you if you where admitted. Responding in such a manner signifies a lack of personal investment on their part, and so I am not surprised it has been difficult getting a follow-up response.
Instead of emailing specific professors, you might try emailing the graduate or admissions coordinator at each of the schools you applied to. These individuals are typically more responsive than individual professors might be.
Looking towards next year, I think you have already identified the ways you can improve your application. The GRE and TOEFL scores must come up. Often applications whose standardized test scores fall below a threshold are tossed aside without consideration for the remainder of the application. At a minimum the goal is to not be in that category. Recommendation letters are tricky because of the time it takes to build a rapport with someone. The best bet is to keep working intensely with your current advisors and possibly branch out to a new lab. In general, a post-doc signing a letter of rec is not highly regarded. Instead you might want to have the post-doc write the letter because they know your more intimately, but have the PI of the lab sign the letter.
Good luck with admissions next cycle.

I gotta step in here and speak out for our OP.
If you're not an international applicant, you have no idea how hard it can be getting LoRs that measure up to the American enthusiastic tone. You might think that every serious academic all over the world is probably comfortable with writing an English LoR, but that's simply not true. It's a sh*t show to apply as an international student and while there is tons of support and expertise available for undergrad students within the US, nobody has given me any piece of advice and my first application failed mainly because of lack of advice and support from my home university. And I come from a German top university. 
Give him a break. This forum is the exact right place for asking what went wrong. Whom else is an international student supposed to ask? 
So, itwlhf: Do not give up! Reapply next year and in the meantime work on:
1) Your Quantitative score in the GRE
2) Your TOEFL score
3) Your Letters of Recommendation. 
Visit theprofessorisin.com and read everything on it. Ask for help. Let colleagues take a look at your application and appreciate critical feedback. And use this forum and all the expertise that is on here.

I don't know anything about your field, and how the application process works in it. But I am an international applicant who has been accepted into 2 of the 4 places I applied to (still awaiting 1 result).
My experience has been that it is almost completely dependent on the nature of the POIs you have been in touch with before/during the application process and how the interaction has been. If that went well, and the person remembers you (or he/she is a genuinely nice person), there's a chance that he/she may respond to your email and let you know what happened. Otherwise, it looks difficult. Even with just 4 applications, the responses have been right across the spectrum, from super helpful and nice to a slightly terse response (I actually got in there!), to complete silence.
While it is definitely not rude to ask for feedback on your application, you shouldn't take a stance that you deserve to know because you spent time and money on it. I don't want to be rude or unhelpful, I am just trying to explain how things are likely to work - please don't take this the wrong way! Think of it from the admissions committee's perspective - it would be unreasonable to expect them to respond to hundreds (or even dozens) of queries on how a specific application is weak.