Terraria – Is it possible to join a modded Terraria game without downloading mods

It is certainly possible. You both just need to have the same mods active.

However, you say you're using the Obsidian mod, and from what I remember there is a part of that mod that is strictly single-player only (specifically, the Parallel Worlds pack), so it may not work as you want it to. Otherwise, running mods in multiplayer is fine.

I don't believe there is. Terraria is solely run from Steam, and the only possible way you could run it outside of Steam was if it was Pirated (which is illegal.) You can still play it if you're not connected to the internet, however. You simply run Steam in offline mode.