What are the game modes for Brink and is it going to be a beta or demo release?

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Dunno yet about modes really.
No open Beta.
Dunno about a demo.

There have been no announcements of a public beta or a game demo.

As for game modes, Brink doesn't rely on DM and CTF but instead have objective-based gameplay, although some objectives will be similar to familiar game modes. Brink does have a few ways of playing through with multiplayer, if I remember correctly you can run through the campaign (as co-op or 8v8), you can play an individual map, or you can play a map twice with the sides switching for the second go-through. I'm a bit fuzzy on this however so take this with a grain of salt (unless I've already been ninjaed )

You may want to read the official FAQ pinned to the top of this forum, and if you have other questions I'd suggest doing a quick search first to see if the topic has come up before.

Iirc a demo has been announced (or mentioned, rather) to be very unlikely.

Campaign mode, freeplay and stopwatch. Campaign plays the story in order, much like singleplayer. Freeplay is the story in random order. Stopwatch is about completing the same objectives faster than the other team, so you switch sides an play the map both as attackers as well as defenders.