If one player doesn't own a DLC-civilization, can the others still use it in multiplayer?

All players are required to have the DLC according to this Steam forum discussion. I believe DLC wasn't usable at all for multiplayer until a patch early 2011; see this discussion.

Unfortunately, you cant use it. You can have it activated in the multiplayer game, but the moment that you try to use one of those civilizations, and someone else on your game doesn't have that DLC, the game won't be able to launch.

Like Sentry said, in the game setup, the host can choose which DLC are activated in a multiplayer game.

As another note, if you try to join a public game, there is a column that shows which expansion packs and DLC are active for that specific game, so there won't be any confusion later. Since private games, are invite-only, you don't get the luxury of seeing that before joining the game, so I suggest that you communicate with the other players before hand.

A DLC can only be used if all players own it.

But you can then simply deactivate that DLC for this game and still play together.