It sounds like your having more then one issue that way look like one problem even though it may be many.
I want to start off by answering your question on the protocol fall back. That isn't exactly how it works. http, or https are functions used by your browser, and the website in question to deliver the information to you. The only difference between http, and https is the security more sespficily the encryption of the traffic while using the site. It can be more complicated then that, but just wanted to help clarify this first. Last point here is the website in question dose make the choice to use one or the other not both.
As far as the issue you are likely having it sounds like you may have some corrupted temp files causing certain sites that originally downloaded them for you to view the site not to display properly, or at all. You should try running a disk clean up on a windows machine. Also go ahead, and delete all cashes from your web browser including those temp files. You may also want to reset your browser to its default settings as well to ensure its operating properly. It may not hurt to update your browser too while your at it.
There is also a good chance you need to call your ISP as they may have some configurations that got messed up on their end.
I also wanted to ask how comfortable are you with working with your modam, or router? Wanted to ask before I go any further plus I think may be best to try these things first.