Gain land by pressing claims of others

Answering as best I can without knowing the full mechanics of succession and exactly how its applied to empires.

In General…

Each kingdom can have it's own succession laws, even if you control more than one - so you can have gavelkind in one kingdom you own and primogeniture for another. A difference in succession laws can mean that kingdoms end up with different heirs, so realms can fracture. What that happens, a game will continue with the heir of the primary title, with whatever other titles he (or she) ends up with.

Because of the above, succession for each kingdom is evaluated independently. The upshot of this is if your ruler has multiple kingdom titles but no Empire title, and all kingdoms have gavelkind succession, you will not lose control any of those kingdoms - since for each kingdom, the title of King will go to the primary heir. However if your ruler had any empire titles (e.g. you managed to become head of the HRE), I think his succession would be handled for the empire as a whole, and the kingdoms owned by the Emperor would go to different sons (or at least I'm pretty sure the de jure kingdoms would - I'm not 100% sure about this part).

In Your Specific Case…

In your case where you are vassal of the HRE, your position is less clear cut. Your kingdoms that are de jure part of the HRE should be bound by HRE succession laws - I'm not sure whether your other kingdoms will be similarly restricted so it might be worth checking. However since you say you have Agnatic-Cognatic Gavelkind in each kingdom, and you don't hold the HRE title yourself, each kingdom's succession should be evaluated independently with the result that each kingdom title goes to your primary heir, while vassals within those kingdoms will be distributed amongst all heirs as you'd expect.

As to why you get the succession warning when Germany is your primary title but not for Italy - I don't know for sure, but perhaps the game only warns you of titles you'll lose within your primary kingdom?

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.)