Does it work without an internet connection?

Yes. Although, you obviously will not be able to play multiplayer games without an internet connection.

Possible to transfer save files?

Yes, but you would need an internet connection. Saves can be stored in the cloud, then transferred over.

Can you install and play DLCs?

Yes. However, they may be issues with certain games if they are no longer on the marketplace. For example, Forza Horizon 1 was delisted not too long ago. I'm not completely sure if the DLC can still be purchased.

Can you unlock 360 achievements you missed on the One?

Yes. When you play BC games on the Xbox One, it is as if you are just playing an Xbox 360.

How often are new games added to the BC library?

Usually a couple games per month.

And are there any other things worth knowing?

Here is a list of all the games that are BC:

http://www.xbox.com/xbox-one/backward-compatibility/available-games