Theoretically speaking, I wish to make an honest buck by walking around and being useless when questioned. Who would I speak to for this? Balmora is a house Hlaalu place, after all. Would it be paid? Would I need a certain rank to do this?

The game doesn't allow you to become a guard straight out of the box......I don't know if there are mods that allow it....but maybe?

If a mod allowed me to do this, who would I speak to?

Oh, I dunno. Vivec maybe ....but he might not speak back.....he's kinda stuck up that way........
I DON'T KNOW....if there was such a mod...it would be whoever the creator decided for you to talk to.
Just put on a full Bonemold suit and stand around and PRETEND.....there aren't many lawbreakers though.....except for the Player character...which is you.......so you would need to clone yourself so that you could arrest yourself.......

Just PRETEND that you are a guard and become a corrupt guard and just steal from the other guards....they have some goodies up in a couple of the towers.

So, anyways... here's a real answer...

You cannot just play a Guard as a normal Class, since Guards are not a Class that can be used by the Player. And even if you do use the Guard Class (via mods/console) that does not mean that you have any affiliation with any of the Guards in-game (except perhaps as a Faction Reputation boost/drop.) The Guard Class has very specific behavior for NPCs, but they mean nothing if the Player uses that Class.

Besides, which Guard faction do you mean? Imperial, Ordinator, Hlaalu, Telvanni, or Redoran?

Anyways, as far as I know, no one has made a mod where you can join a vanilla Guard faction for any of the standard Houses.
The closest you're going to get is by joining the Imperial Legion or Temple and using your imagination.

"So, anyways... here's a real answer..."

Being influenced by PorridgeIsGood's signature I was beginning to wonder if it were a real question......

So, anyways... you gave a really good answer. Imperial Legion is as close as you're going to get....talk to Darius in Gnisis....You will find Darius on the lower floor of the Madach Trade House in Gnisis.
....but I see over in the mods forum somebody is kicking around a mod idea of becoming a member of an Ordinator faction. PorridgeIsGood, you might want to follow that one.

Perhaps I should've placed this in the lore forum.
Had Morrowind been even richer in content, how would I, random dude, become a Hlaalu Guard for the city of Balmora?

When I did it, I joined the legion and used the Morrowind Comes Alive mod, which causes random enemy spawns in cities, so I could feel liek was occasionally taking down a criminal while on patrol.

You'd have to 'cheat' in your Paycheck.

I do this every 10 days on my characters. Keeping track of 7 days is kind of taxing. But when I see a x0, I know it's payday.

Really exciting. A guard in Seyda Neen or Caldera or other podunks. Whatever...

If you're going in terms of Roleplay, I assume you'd go up to the Hlaalu councilhouse, join the house, rise to the rank of Lawman, and ask to defend the city of Balmora as a guardsman. (I assume all the guards are lawman rank, I could be wrong)