Computer related

[mtread]

Your Web page editor will be producing Dynamic Html, otherwise known as Javascript to do these clever things like drop down menus. Html on it's own is not capable of this. It may well create CSS style sheets as well to keep appearance consistent.
All browsers 'parse' these web pages and interpret the combined Html and Javascript instructions. The way they read Html is fairly consistent, but not necessarily so with Javascript. Javascript is routine based code, like any programming language, whereas Html is just a set of tagged commands read in strict order.
Browsers set to 'safe mode' will almost certainly switch Javascript off to counteract malicious code, but that is also why the drop downs don't work when you switch to safe.
So my guess is the problem is caused by a combination of :
The way your editor produces the Javascript,  and
Your browsers settings
I'm also guessing, given that you can sporadically see them, something on your pc is changing your browser settings back and forth.
The good news is that it's not something on the hosting  server side, as we can all see the pages fine. It's only you that can't see your Web pages correctly :-)