My login doesn't last, and site is still unsafe.

[Gambit37]

Things are still weird here regarding logins and security:

• Recently, I've had to login again every day. It seems like the login cookie is not persisting.
• Chrome frequently blocks me from entering the site with its standard "Site is unsafe" warning page. I have to click through the Advanced button on that page to get to the site.

I think there's something wrong with the site's certificate and/or the way the secure and unsecure web addresses are linked. This stuff was always confusing to me when I used to admin the site, but I don't recall such problems on the host we used years ago. I wonder if there's something misconfigured with the current host? Anything I can do to help with that?