Support

First of all, what you are asking has nothing to do with Admin Tools. That makes it outside our support scope.

What you describe is the intended behavior of Joomla!. It makes no sense logging in through HTTPS and then switching back to unencrypted HTTP. This would make it possible for hackers to perform a session hijacking attack by stealing the insecure, unencrypted cookies.

Since early 2011 Google has dispelled the myth that HTTPS is computationally expensive. They have said that the impact of HTTPS to the server load was between 1% and 2% which is completely insignificant. As a result the advice for the past 5 years has been to go full HTTPS on your site. In fact, the new HTTP protocol (HTTP/2) doesn't even support unencrypted communications any more.

Considering that you are making this support request to a company publishing web security software, our advice is to enable HTTPS throughout your site by default. You can do so by first going to the Global Configuration of your site and setting Force SSL to Entire Site. Then, if you are using Admin Tools' .htaccess Maker, make sure you enable the HSTS Header option (it tells the browser to never bother contacting your site through unencrypted HTTP).

Regarding the missing CSS, I suspect that you are referencing external fonts with plain HTTP URLs, i.e. URLs that start with http:// instead of https:// (HTTPS) or // (let the browser decide). When you are visiting the site over HTTPS these are considered "insecure content" and will either cause a scary warning dialog to appear (older browsers) or will be silently ignored (newer browsers). Audit your CSS and adjust the URLs of external resources.