1) Javascript void(0) is normal to see in the link hint for where a click will take you. The link is actually executing a bunch of Javascript that will log the user in, when things are working properly. I just tested on your site though, and the Facebook authentication popup did appear as expected. I didn't go through the reg process though to fully test.
2) I'm a bit unsure what you mean here. Are you talking about Like'ing things as your Facebook Page? If so, unfortunately, you can't. Facebook only allows personal user profiles to Like content. Since a user can create unlimited Pages, it would be abused very quickly to get unlimited Like's for something. If you meant something else, please help me understand what mnyname/business and myname are (Personal Facebook Profiles, Facebook Pages, etc).
3) This is almost definitely the cause of your any authentication problems you're having in #1 above. The code in question is this block, directly below our Facebook Javascript inclusion after your opening body tag:
<script>
window.fbAsyncInit = function() {
FB.init({
appId : '1471593899787359',
xfbml : true,
version : 'v2.1'
});
};
(function(d, s, id){
var js, fjs = d.getElementsByTagName(s)[0];
if (d.getElementById(id)) {return;}
js = d.createElement(s); js.id = id;
js.src = "//connect.facebook.net/en_US/sdk.js";
fjs.parentNode.insertBefore(js, fjs);
}(document, 'script', 'facebook-jssdk'));
</script>That will need to be removed for Facebook authentication to work properly. I'd look in your System plugin group to see if there are any other plugins related to social networking.
I hope that helps, but if you need anything else, just let me know.
Thanks,
Alex