I already selected which wargoal I want to add, and I if didn't have enough jingoism, it'd imho make "Add Wargal button unpressable in the first place.
Justified wargoals are addable anytime. If you've got a justified wargoal against your opponent, but not enough jingoism to add unjustified ones, 'Proceed' is greyed out for any and all unjustified wargoals.
When rebels takeover, do you lose puppets like you lose spherelings?
Political Rebels, yes. Liberation Rebels, no (spherelings remain too).
Do you get militancy when an event causes a ruling-party change?
Not directly. However pop's who not agree with the ruling party get a daily militancy for that fact.
How do I get Austria to join my German Empire? Year is 1880, I have austria sphered, and hungary exists like it says in the wiki. HOD.
Waiting. The mechanism is event driven, so it can happen tomorrow or - extremly rare, though - never.
Anyway to force nations to giveup their cores in a peacedeal?
Nope. Liberate Country kills cores, though.
Or if you take a core from another nation you will have that nation declare war on you like a clockwerk every 5 years?
Usually yes. Uncivs almost never dow civs, regardless of what reason they might have. Civs will dow you every ~5 years, unless you're multiple times stronger (alliances are considered to some extent).
I don't think so, but I haven't seen any event that actually does that.
There some (rare) event's in vanilla, while mods like APD build parts of their political model on that mechanism.
Again, I don't think so, but there's nothing stopping your puppets from going "hmm... maybe now is a good time for independence?"
Puppets get killed by political rebels. The two most frequent examples are Luxembourg (not to confuse with the salic law event for Queen Wilhelmina 1890) and China.