As for your first issue of joining a random server and being greeted with the "missing required dlc" message, it is likely that you are attempting to join a server running one or more of the paid DLC packs (i.e. China Rising). To solve this you can either buy the DLCs one at a time or by purchasing the Premium Pass for Battlefield 4.
To solve the second issue I suggest that you redownload Night Operations and then perform a Game Repair on Battlefield 4. (Rightclick Battlefield 4 in Origin > Repair Game...)
