After that I tested my .htaccess line by line and found what causes Apache to serve malformed and non-existing URL-s. The following line
AddType x-mapp-php5 .php .html .htm .shtml
An extended search gave no results to solve the problem. If I remove the line all is ok, but I really need it there since I need PHP for serving Google adds in my website and the only way to have it on 1and1 shared hosting is by that line (I got it from 1and1 support).
I also found other URL-s ending on .html/some characters which SE are indexing as pages from my website.
The only solution at the moment for me seems to be: to rewrite all bad url-s to correct ones i.e.
to rewrite all ending on