Disappearing dock and menu bar only on other desktops

There is an app called Bartender, it's paid through. As far as I remember it has a full functional trial and a lot of useful functions, so you should try it.

It sounds like what you are describing is what happens when you go in full screen mode in any app, the menu bar and the dock hide regardless of the dock auto-hide settings. I would suggest just resizing the safari window instead of putting it in full screen mode. (If that is what is happening.)