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?
The strong claim does not become weak, but the strong claim can only distribute to first 3 children, the remaining would get a weak claim. You can have more information from here.
You can get claim via marriage. Marry someone who has an inheritable claim. Your children will have the claim.
You can also get de jure ducal claim by creating/unsurping duchy title although the claim does not appear in the claim list. The same goes for Kingdom and Empire.
Invite pretenders into your court, and then grant him landed title.
Visit the wiki for more information.
Best Answer
The easiest way to check sucession laws for a title is to find the owner of that title and mouse over it on their character sheet. This will show you the law for that title, the heir, and the line of succession, i.e. who you need to get rid of to inherit it.
For other laws, select the De Jure Kingdoms view and mouse over the area. This will tell you laws such as regulated inheritance, or how bishops are selected. Note that these laws are based the De Jure Kingdom that county/duchy is in, if it exists, not necessarily the kingdom of its owner.