This sort of "protests" can be useful in granting political capital to this or that politician to come out in favour of (or refrain from adopting) this or that position, eventually snowballing into official government policy.
Yes, that's the purpose of a demonstration (it's a way to demonstrate the political power lent by the people to an opinion). However, I just don't see how this would change anything in the current situation since the incentives of avoiding a war already seems the predominant objective.
I wouldn't be so sure. Drawing the Russian military in a lengthy and expensive campaign to pacify a huge country is actually a very compelling scenario from a certain point of view. Yes, Europe would be short on gas for a while, but quoting one Ms. Nuland, "F*ck the EU", right?
There are a lot of incentives on all sides, at this point.