No, they don't. In fact, I'm assuming you purchased the key via EA, which means you won't use Steam. If you purchased the game via Steam, you would download it via Steam.
Regardless of whether you purchased the game via Steam, via EA or from a retail store, when you connect to the game servers, that has nothing to do with Steam. It is independent of Steam or EA Manager.
For example, I purchased my copy at Best Buy because they had it on sale for $30. When I installed it, I didn't use EA Manager since I have the actual DVD medium. However, I did add the game to Steam so that I can launch it via Steam if I want to, but I don't have to use Steam to launch it. I can go straight to the game icon that was installed via the installer and launch it that way, never using Steam nor logging into Steam to launch the game.
If you purchased the game via Steam, Steam can be used to launch the game, but once you go online to play multiplayer, it's the same for you as it is for me. The Steam server is NOT hosting the game. That is a totally separate server away from EA and Steam.