If there is a decent break angle over the nut and the open string still sounds dull compared to when it is fretted, then I would look for the problem in the nut. Be sure that the nut slot is tilted slightly downward toward the headstock, and that there isn't side-to-side movement of the string in the slot.
A little bit of update to my problem:
I ordered new string set ( GHS boomers Flea signature ).
After that, i checked the break angle over the nut. Well, the couple of things that i noticed: I had almost half of string wind on the E string tuner ( i mean, 0.5 wind. it was not even winded properly! ). I dont know how i missed that important point. Well, you learn something new everyday!
Also, there was almost NO angle from the nut to tuner, since the string was not winded properly. Also, i filed the nut slot a little bit to the tuner side.
Installed new strings: now i have 3-4 winds on E string, and at least 3 winds on A,D ang G strings. The angle is very smooth, at least, you can see it now
The rattle and buzz from the nut is gone! ( yeah, i didnt mention the buzz in previous post, as i thought it was combined problem with the dull - sounding E string.)
Also, the open E string and 12th fret on E string sounds as good, as any other fret and note - the problem is solved!
P.S: taking off the old strings, i noticed that they are not straight at all, when hanged in the air - the strings wind themselves in spiral from start to bottom. My guess that they were also twisted...