>>> Currently I prefer to concentrate on the case of accessing X selection 
>>> or getting DISPLAY without creation of visible frames.
>> 
>> Sorry, I don't understand what "getting DISPLAY without creation of 
>> visible frames" means.  Is the $DISPLAY environment variable not 
>> enough?
>
> I do not see a way to get the value of the DISPLAY environment variable 
> specific to the emacsclient process.
>

But where do you need that value, and what do you mean by "without 
creation of visible frames"?  Not all emacsclient processes have a 
corresponding display, e.g. if you just do

emacsclient --eval '(+ 1 1)'

there is no display involved.  If you create an emacsclient process with 
-c or -r, then in Elisp you can get the display on which the frame is 
displayed with (getenv "DISPLAY"), or (cdr (assq 'display 
(frame-parameters))).