OK, so the TOC is what's broken. As always guys you are correct

So what I did: in Convert books menu in Table of Contents I chose 'Force use of auto-generated...'. It actually helped with stats working properly BUT I lost all the subchapters information completely.
So I tried option 'Insert inline Table of Contents' option in Kepub Output and it gave me pretty nice ToC. Can i somehow force calibre to put it as main ToC file? By default it only attaches it to the book in the beginning or at the end. Below are the shots of original (left) and generated by Kepub Output (right) TOCs.
PS Standard conversion to Kepub doesn't update ToC in a way that would help.
PS2 OK, now I see that there is a toc.ncx file as well, not only xhtml one. I have no idea what I'm doing