Or there could be legitimate discontent with how horribly things are going.
Germany has massively risen in power, both political and economical. This is partially due to political mismanagement/corruption (esp. Eastern Europe), but a huge part of the growth of Germany results out of the loss that has been paid in poorer Europe.
And, surprise, both foreign countries and the German left do not like this.
That, and conflict in which aggressor aimed to physically destroy our population followed by half of century of being colony for USSR also had its part in our economies. When 90's finally came we had few people that remembered democracy and even less businesses that could kickstart our economies, only massive debts inherited from communist goverments.
I suspect that on a certain animalistic level, the "message" that the protestors want to get across (to Merkel and people at her level) is, basically: "Do what you have to do. But don't pretend to feel that there's anything dignified and civilized about it."
It's just that states only view their own use of violence as legitimate.
2. You make it sound like there wasn't overwhelming support for delegitimizing use of violence by any unsanctioned actor. That implicit agreement is what enables society in the first place.
They're correct. That is the entire premise of the "monopoly on violence" which defines a state.
You can possibly argue that violence is effective, but by definition it's not legitimate, unless you choose to simply disregard the framework by which that legitimacy is conferred and substitute your own, in which case your criteria become relevant to no one but yourself.
However, it does not follow that the state is not using violence to enable speech.
State violence is still violence. The army of riot police descending on the city is there for a reason - to suppress the rights to freedom of speech and assembly for the many.