Terraria – Does Corruption/Crimson/Hallow Spread on a Multiplayer Server when no-one is logged on

With the Clentaminator present in the game, this is actually quite easy, if a bit time consuming and expensive. Dig hellevators every 120 blocks across the area you want to purify. With the Green Solution equipped, float down the each hellevator twice, first spraying to the left, then to the right. I recently did this in one of my worlds, and it worked perfectly. I'd recommend checking every so often after you're done to make sure you didn't miss anything. This will also turn any underground mushroom biome to jungle, so you may want to go back when you're done and convert it back with Dark Blue Solution.

When I did this, I left large swathes of land from the surface to the underworld converted, and even brought in Crimson from another world so I had all the biomes present. Just make sure that if you do this you put wider shafts (I left six blocks) around the corrupted/crimsoned/hallowed areas so they don't spread.

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.