I’ve spent all night trying different settings and checking if it works or not. Figured out that the problem (at least with my website) was with the Page Cache Debug turned on (WP Admin -> Performance -> General Settings, Debug section, Debug Mode: Page Cache ‘checked’). When turned off, cleared and rebuilt all caches – seems to work fine. Pages are cached, everything is working now for both logged in users and not, in different browsers etc. Tested via tools.pingdom.com, Google Page Speed and numerous other resources – all of them worked and showed good rating for my site. Before, when I had blank screen, tools.pingdom.com (and others) was loading only this blank white page. Now it seems to work. Will ask some friends to check the site tomorrow to see if someone will get the white screen. I hope not ??
BTW, had to turn off CDN as well, because all performance analyzing tools were telling me to minimize redirects from a CNAME mapped subdomain (media.mydomain.com) to Amazon S3 bucket (media.mydomain.com.s3.amazonaws.com). I thought mapping a subdomain to S3 bucket will not cause any redirects ’cause it goes on a DNS level. Seems like I was wrong. Anyway, for this particular website 98% of visitors are from Russia and Ukraine, server is located in Russia, so it serves static files to local visitors much faster than Amazon S3 – 25ms vs 150-200ms ping makes the difference.