Milko,
1) Not sure why Autotune is returning that error and not showing your site information on the overview page. That's actually not a huge deal since it seems like everything is setup correctly looking at your Autotune Settings and the fact that I can register/login to your site. So, we'll save that issue for later to try and fix the other problems first...
2) Can you try changing the "Facebook Language Locale Override:" in the Configuration -> Facebook API section of JFBConnect to "en_GB" and see if hovering over the Login With Facebook button changes the SSL message at the top still? If not, that would be a Facebook bug. We can submit it for you, but would like you to verify. We've never seen that before and it doesn't happen on this site (or others) we've tested.
3) Opera is very strict as far as standards and Javascript goes. Looking at your page, below the closing body tag at the very bottom, CometChat is adding more code. That violates HTML standards and since it's adding Javascript, that could easily be causing Opera to have issues. Also, do you have problems on this site with Opera? That will help narrow down the cause.
Additionally, when I try to login on your site, in the Opera Error Console is the following message:
Opera has modified the JavaScript on www.facebook.com (facebook: work around Opera's too strict origin checks on https for https -> http(s) other site -> h...). See browser.js for details User Javascript thread
Opera has Facebook specific code to work around some integration issues with their browser and Facebook authentication. At times, this means that when Facebook updates their Javascript library (which happens pretty often), it breaks some features on Opera until they update their workarounds. Unfortunately, there's not anything we can do to fix these issues as that's an Opera/Facebook issue and not something we can work-around.
4) Finally, I'm not sure why you can't delete posts. If you disable the JFBCSystem plugin, does that allow you to delete posts? If you change the JomSocial template, does that change the behavior? Do you have the JFBConnect - JomSocial Post To Wall plugin enabled? If so, try disabling it.
We tested on a JomSocial 2.6 site and couldn't recreate your problem, and haven't heard of it before, so it's definitely strange. It's also possible the extra Cometchat code at the bottom of your HTML is interfering as well.
Hope that all helps get you started,
Alex