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?
Yes, other non-descendents can inherit.
Think of it this way: they are a descendent of the previous holder of the title (the current ruler's parent), so they are already in line for the title. One brother inheriting doesn't remove the other's claim to the title, so the second brother is the first brother's heir, at least until the first brother has children. Then the second brother is still in line for the title, after the first brother's children.
Best Answer
Problem: once you die, your sons will split up your lands. Since you have more than one top-level title (duchy, in this case), your multiple sons will split into multiple independent realms.
Solution 1: MURDER. You have too many sons. Make terrible things happen to all but one of your sons. Cross your fingers and hope your sole surviving son doesn't die until after you do. ("Terrible things" include assassination, putting them at the head of a far-too-small army and sending them to war, making them the court chaplain and sending them to proselytize to the pagans, and rarely-available plots to kill.)
Solution 2: TREASON. If you only have one top-level title, then your oldest son gets that title, and your other sons will end up as vassals to your oldest son. It's time to become King of France. You'll need either a majority of the counties in France, or a claim on the French crown, or permission from the Pope to invade France (which will probably involve excommunicating the French king first). But before you can take that final step to acquire the French throne, you'll need to be independent from France itself; this can be made much easier if several other dukes are attempting to declare independence at the same time. (Try joining or starting the Faction for Independence?)
Note that your lands will still be split up among your sons, but at least your eldest son will be your other sons' liege. Your eldest son probably won't have a full demesne after the inheritance kicks off, but maybe you can pick up some counties elsewhere, via holy wars or pressing claims.
Solution 3: WAR. Your sons will end up as independent nobles from each other (as vassals of the King of France), but they'll all have strong claims on each other. Your eldest son will end up with all your money and the best lands, so have him go to war to reclaim the rest of your former lands! Make sure he gets enough money to hire mercenaries, and it will be even easier.
Note that if your liege has Medium Crown Authority or higher, you won't be able to do this. At Medium authority, wars between vassals are prohibited.
Solution 4: ....Peaceful discussion? Once per generation, you may change your inheritance laws. If all of your immediate vassals have a positive opinion of you, and they aren't at war with each other, and you've ruled for at least 10 years, then just change your inheritance law to something else. Primogeniture requires High crown authority, but you should always be able to get Elective. (Elective has its own troubles, but you should be okay for at least this one generation.) Primogeniture is the easiest succession law to control, but your pretenders will dislike you (a net -50 opinion penalty!), and you'll lose the +5 opinion bonus you had with your other vassals, but it's (usually) better than splitting up your lands. Also, switching out of gavelkind will reduce your demesne size a bit.
For more details, wander through the succession and gavelkind sections of the CK2 wiki.