Like, during the Mecklenburg war, I never took part apparently. Despite my forces being at the Kaiser's command. How does that work exactly? How could I have joined the war appropriately? I did raise my levies, led them off to fight, and suddenly found the 4k Mecklenburgers in my path rather than them defending their homeland. So my forces were wrecked and all that stuff. Luckily no one bothered me.
'Kay so, your Empire went to war with Mecklenburg, which means the Kaiser is fighting them. You are not. You are however a vassal, so you are considered to be "at war" with them, which means they consider your forces to be part of the Empire, and they WILL fight YOU if you happen to get in the way. And since they are at war with the actual ruler they also get the chance to besiege and take your lands in order to increase their warscore. They aren't actually gunning for you in particular though, they are going after your boss.
Now, since Bohemia is so far away from Mecklenburg you really should just ignore the war. You are not fighting them, let the Kaiser do that. The more troops he has to throw into the meatgrinder the weaker he becomes relative to you (and his other vassals). In turn that means he will be less able to centralize authority and make you actually listen to him. Furthermore, you are not expected to help beyond the soldiers you actually send when he raises the levies. Your own personal troops do not need to be deployed.
At some point, there was some legislation article that popped up saying I could vote on the Limited Crown Authority - but I didn't know how.
If you get a vote then your options will be in that message. You only get a vote if both of these are true: a. the Council GETS a vote on these matters, and b. you are on the Council. Otherwise the Kaiser isn't asking, he is informing you that it has happened.
The Kaiser also raised some of my levies, taking my Duke, Chancellor, and Martial off to fight in Mecklenburg.
I haven't played the vanilla in...years actually, but you might be able to click on a button on each of those characters to forbid them being able to lead armies.
As for general points, early in the game there really isn't that much to do. You need to micromanage construction and use your council to generate extra money/tech points so that you can start to build advantages, but you likely won't be warring often. As the Duke of Bohemia your first priorities should probably be murdering all (or at least several of) the counts who are under you, and their families. That way you inherit their castles, and can increase your own income, and your levies. You could also imprison them and revoke their titles, but there's less murder in that plan so I'm not sure why you would.
This will have the side effect of having any vassals you have who are left hate your guts, but if you control all the counties in the duchy it really doesn't matter. From there find a weak neighbor (normally a count) and try to force them to become your vassal (or if you aren't at the upper limit on counties just make war upon them and take it for yourself).
To do this you need a claim. You chancellor is the one who will form it, the middle button of their missions is the "Fabricate Claims" mission, and they have a percentage to complete their work every month. The fastest is about 20% per year. So it can take a while for them to finish. You will have to pay for that claim when it is fabricated out of your personal wealth, so be sure to keep that number relatively high.
Also, be sure that the weak neighbor you are bullying has the same direct superior as you (in your case the Kaiser.) Otherwise you will have to go up the chain and fight the vassal who DOES have your direct superior above them. Which might mean you are fighting another Duke. Bullying counts is much easier, especially early in the game.