My ledger says I have 19 direct vassals but the upper-right corner says I have 10 vassals

Here's how it works: If you are their liege, the gains from the war are part of your kingdom. It doesn't matter if they're your king, duke, earl, or even mayor, bishop, or count.

To claim a duchy from somebody else through war (when you don't have a casus belli that lets you claim it directly):

1. Find out who* has a claim on it by opening the duchy information screen and clicking on "Claimants" to see who has a claim on that kingdom.

2. Invite one of the male claimants over to your court. (If nobody will come over, or there are no male heirs, you'll have to find another way to get that duchy. Note that if one of your courtiers marries a female claimant that you can't legally give a title to, that female claimant is not your vassal and the newly-conquered duchy will not become part of your kingdom, even if you give your male courtier a title.)

3. Give that claimant a landed title. Any landed title will do, even a city, church, or castle.

   Note that if you land the claimant a city, gaining him a Duchy will found a vassal merchant republic instead of gaining you a new feudal vassal if you are playing version 1.9 of the game or later. Merchant republics are generally considered awesome to have as vassals due to their revenue and tax bonuses, but this might disrupt an existing merchant republic that is already earning you revenue.

4. Go to war with the ruler of your choice, using the "Claim on duchy X" as your casus belli.

5. Once you win the war, that claimant will be your duke.

I'm guessing that this will also work with kingdoms if you're an emperor. So far, I've claimed two duchies this way: once by giving away a county to the claimant, and once by giving away a bishopric.

*As revealed in this other answer, you can't actually press the claim in war unless the claim is strong, or the claimant is a pretender (2nd or 3rd in line), or there's already another war going on over the title, or the current holder of the title is a woman (and you have a male claimant) or a child (so there's a regency). You can only use "any claimant with a pulse" if the title is disputed or the ruler is a child; you can only use "any male claimant with a pulse" if the ruler is female; otherwise, you'll need to pick up one of the pretenders or someone with a strong claim: you'll need to do slightly more research to find who they are, and they're often harder to invite to your court.

(Note: if there are three or fewer male claimants, then it's almost certain that they're the heir and pretenders. Just grab one and go on your way. Figuring out the exact claims is more important when there are four or more male claimants.)

Clever, you found a way to create a merchant republic I hadn't considered before.

First, I think the only reason you need the soon-to-be Grand Mayor a county before granting a duchy is because the game won't allow you to grant a Baron-tier character a Duchy directly. Apparently you can get around that with claim wars. The city doesn't actually need to be in the county, nor the city or county in the duchy, it just makes for cleaner maps and less arguments between vassals over who should control what.

Second, unless crown laws (Medium or higher) prohibit it, the Grand Mayors are likely to try declaring trade wars on one another and possibly get Feudal Dukes (or even outside powers) to Embargo the other republic. They will also likely be plotting to steal each others trade posts and if the new republic gains enough posts they could threaten the old republics trade zone bonuses. That kind of internal strife costs money (and thus is inefficient), so it's probably better to quietly smother your fledgling republic.

Third, I don't think the game will let you revoke the city if he holds no other county or baron tier titles. You might be able to revoke the city if you grant him a feudal county then strip him of the city title. Of course, then you've created a Duke with a reason to hate you, and the two unjustified revocations will give you a -40 relation penalty with all your vassals. The simpler option, which will offend your vassals less (-20 instead of -40), would be to simply revoke the Duchy directly. The former leader of the republic will go back to being a regular Mayor and you will have a Duchy to hand out as you see fit.