Ok so I'm very new to Morrowind, just installed it on my PC a few weeks ago, and have since installed the "Sound and Graphics Overhaul" mod and a mod which replaces faces on the NPCs.

My question is: Is there a risk of compatibility issues when installing multiple mods? After looking at some of the other mods out there, there are several more I'd like to install, but many seem like they may be redundant in nature. For example, if I install another mod that adds some new NPCs, but also makes the faces of all the pre-existing NPCs different, will that somehow conflict with my other mod that also adds new faces?

I know this is a very noob question, but I'd appreciate any help someone might provide!

Right, clashes depend very much on what you add. To use your example, it depends - not very helpful, but true.

If you add a head pack that adds completely new heads, there shouldn't be a clash. If it's two mods that both alter the stock heads, there will be a clash. This isn't fatal though - what happens is the last mod you loaded will take priority and be the one that is used. So, if both mods alter head A, it'll be the mod that you paste the file in for last that is used. This is because the files for that head will be overwritten when you add the new meshes/skins/whatever to the data files area. If you're talking about one of Westerly's packs, they go fine with everything. I've got mixed feelings about some of his hairs though. It's odd when NPCs all over the game suddenly sprout emo-fringes. Especially the Khaijit.

For other clashes, most of the really big mods offer comparability patches to smooth things out, or at the very least will offer advice in the readme as to what will or won't work with their mod. When it doubt, always consult the readme. If that doesn't help, try to contact the author or open a thread here. Most mods will work together, apart from obvious things like if you use two alternate beginning mods, or two house mods that are in exactly the same place.

Easy but difficult answer

You can install any mod together as long as they do not modify the same object as any other mod.
Yet the outcome is different for every field, like objects, dialogue and quests, texture replacers.
Sometimes you get better effects using two texture replacers on top of eachother, and this is not dangerous, but using two mods that say enhance Tel Uvirith, you must be careful and read the readmes before installing, always do!
Before mods get the most needed, as MCP, and maybe MPP, to patch MW, so it runs better, and it also gives new possibilities for mods.

Ok probably another noob question, but what is MCP and MPP? Thanks!

Morrowind Code Patch (MCP)
Morrowind Patch Project (MPP)

:D

Also I strongly reccomend playing MW and expansions with the necessary patches alone. First finish the game vanilla and experience it, then see what you want to enhance and what not.

Vanilla Morrowind is plainly good!!