Without a doubt, having lots of tabs open can impact a computer’s speed. Whether or not it slows down the entire network will depend on exactly what all those tabs are doing.
Appdata roaming doesn’t mean your computer has been roaming. It’s nothing more than a folder designation on your machine. Why data may be put there, however, is a bit more complex.
The term “server” is used a lot in many contexts, but it’s really just a computer; quite often, it’s one that isn’t all that special.