Reviewing your install right away can see $live_site is populated. This should NEVER be the case. It prevents dynamic generation of live_site so you site is always loading content with HTTP and never with HTTPS. This means you'll always have mixed content.

Please edit configuration.php and set $live_site to '' (blank). This variable should never be populated (any extension suggesting otherwise should not be used). Once done please confirm if still having mixed content issues.

With quotes, without would cause a fatal php error.

it currently has single quotes : 'like this', should I leave it like that or give it double quotes or does it matter.

Single quotes is the recommended. Either way won't matter though.

I'm worried about making this change throwing my site into a tail spin. We are at a critical point with the site and I can not having it going down right now. Do you foresee any issues arising from making this change? I just want to make certain before I do this.

$live_site by default is blank, you had to or something had to set this manually. It should always be blank. Some extensions require $live_site, but can't recall them off top of my head. You'll of course need to run through your site testing features to ensure all working fine.