FreeDoom is an open source game running on the Doom engine, but no, Doom's assets were never open sourced.

Quake III Arena had Open Arena, which was an open source clone. Did anybody ever do anything similar with the original Quake engine?

Both have open source imitations; I think.

You need the wads (where's all the data). If you have doom on steam, you can compile something like GZDoom, and copy the wads from steam over

Doomsday is great (apart from the occasional segfault at the end of a level). It works well with Steam files and it has its own launcher so you can customize things easily.

Of course, but even Romero has released free WADs so content is not an issue for true Doom fanatics.