For the Mass Effect Trilogy, is the DLC content included on the disc, or as a dowloadable code?

Depends on the platform, but the general rule is that it's mostly the same as when each individual game first came out. This means that most DLC is still DLC (that you have to buy), except for Mass Effect 2 on PS3, which came preinstalled with some major DLC missions.

The PC version comes with Bring Down the Sky and Pinnacle Station on the disc for ME1, and on PS3 it comes with Bring Down the Sky for ME1, and Kasumi, Overlord and Lair of the Shadow Broker for ME2 on disc. The rest of the DLC needs to be downloaded as far as I know. :)

It should come with more than one, if I recall. If you lose them, you can always buy the DLC, which are kind of cheap anyway.