Small update after having tried a couple of methods, only one succeeded.
1) First we went the ssh way as you can see in the MediaTemple article Run_PHP_as_FastCGI_and_resolve_Apache_permission_errors
But that didn't work. Maybe the article's outdated (considering the apache directives it uses e.g. 'MaxProcessCount'), or it works only on mt (but I doubt as our client's server is quite straight, nothing special)
Still, would have been handy as I first didn't have access to the Scripts settings in Plesk, but had ssh + root access.
2) Went the Plesk way as iDanny wrote, right after we got the access to the Scripts settings for that domain in Plesk.
All went fine. Site is running 2-3 times faster than before.
As for the domain's php.ini and PHP session path, they can be handled on a domain base (i.e. no need to edit /etc/php.ini and to put session.path in apache /tmp).
- Plesk handled the domain's php.ini file after we told it to do as quoted in the Paralleles KB's guide bellow.
- We created PHP session path in <HTTPROOT>/../var/tmp: works fine. Then in the <HTTPROOT>/../private directory: works fine too, as shown in this post and parallels' guide:fCGI var/lib/php/session Permission Issue(Parallels KB for Plesk 10.4) PHP on domain running via CGI/FastCGI can't use session by default
Last thing, markusromanus there's no /etc/apache2/mods-enabled/fcgid.conf (everything is under /etc/httpd). Maybe it's a distro relative place.
Anyway it's running fine, installation (and update) of >1MB extension included without that line (FcgidMaxRequestLen) atm.
To all thank you very much for sharing the tip and experiences.