Ok so... talk about getting myself into a mess lol.
I went ahead and ensured all extensions needed on a typical php setup are working for the Zend Framework and phpQuery on shared and managed hosting.
For the most part the problem is resolved but out of countless activations and initializations it seems to work and maybe fail once or twice. So it may be a random issue on shared hosts (managed hosts works so far).
However, to my rotten luck, the initialization does not work on my local development server at all. I don't know if I did something wrong or I have successfully replicated the bug - (permanently? :) ). This happened after I made sure that all typical php extensions are installed and working on this server.
Everything else does work in AAM just not the initialization. One thing I can tell you Vasyl, is that Json returns status "success" for every post type. I double checked to ensure that it's not returning "success" with no reason by installing bbPress. Json now returned "success" for bbPress post types also... just not showing anything.
So if it wasn't something I did wrong then it must be happening with phpQuery or at least ajax that it's not returning the results.
I hope this helps in any way. (Btw, every test I made so far is on a fresh install of WP).