To answer your questions in order:
1) Mixed content. The typical StackExchange page combines content from a number of sources (ad servers, imgur, gravatar, etc.), and not all of these support HTTPS yet. Browsers react in varying ways to pages that contain a mix of HTTP and HTTPS content, but the most common is to refuse to load the insecure content, leaving you with a broken page.
2) Yes, it's a security issue. It's not as much of one as it could be, though, since the login pages use SSL.
3) Maybe. It depends on if StackExchange is using session fixation techniques to prevent session stealing or not.