Right, it works now and the reason is - I changed my hosting provider

. No, this decision was not caused by non-working feeds

but by the number of outages they had in their network. I just got tired seeing my site down again and again. And btw this hosting provider is #1 in top 10 hosting providers' list in the USA. Their 24/7 support was useless, so whatever I describe below is based on my own attempts to fix the issue with limited access to server configuration. Here is what I found and tried to do:
Empty feeds problem was observed only with RSS feeds from feeds.feedburner.com. By saying "only" here I obviously mean only among the number of RSS feeds from different sources that I tested.
After searching Google I found few posting in different forums saying that feedburner.com requires certain additional information from the client to provide non-empty reply. They were mentioning User-agent field and I tried to change php.ini to embed it, but that still did not help.
Then as a workaround I added wget into cronjob on my home Linux box that is provisioned in DynDNS and can be accessed via FQDN from the Internet. I was saving the feeds locally in xml files and then my main web site was retrieving them from my home web server instead of going to feedburner.com. This worked just fine, however, as I've said earlier, all those outages forced me to move my site to another place and I've not been having this problem since then. I got rid of my workaround and everything works as it should.
My conclusion is that certain setting required by feeds.feedburner.com was missing on the server, but unfortunately I did not get chance to find which one.
If someone will stumble into this problem and find the solution, please post it here. Then, hopefully, KL-RSSFEED design team will be able to include it into one of the next releases.
Thanks.