I used to play a decent amount of ArmA and although I never actually ran a server or did any of the scripting for the missions the group I played with I believe I can pretty conclusively say that
You can run other mods as well as the one's required by the mission, unless the host has spesifically told the server that you can only run X, Y and Z mods (might be an option for dedicated servers only - not sure on that). Though running mods that the host isn't running may cause unfortunate errors depending on the mod in question. With weapons you just end up running around with a gun no one else can see or hear, some of the UI mods (like ShackTactical's HUD mod) do mostly nothing and some mods might just end up conflicting with the required mods and cause a crash.
The group I played with had at best (worst) 50 mods running simultaneously with only very minor problems that most of us didn't even notice, but that did take a huge effort in order to test and make sure they weren't conflicting with each other. So you can run many different mods at once, but it does increase the risk of just breaking the game. Like Skyrim players tend to do: mod it, 'till you break it.
So the safest way to go about it is to make sure you only run the mods that the host has loaded themselves (i.e. the required mods).
The ArmA 3 launcher is a good tool and does pretty much everything the mod managers did before Bohemia released the launcher so at least in my mind you don't really need a separate mod manager anymore.
