Note: I am only going to talk about word and phrase meanings, not the actual events your passage describes.

1) A number of words, phrases, and figures of speech are used to imply that a public figure--a head of state, a premier, a prime minister--is not really acting on his own, but is just following the order of some other group that is secretly in control.

In English, the word "puppet" can mean what is more properly called a "marionette." These are little jointed wooden dolls. They act out plays on a miniature stage. They seem to be alive and moving by themselves. In reality, each puppet has strings attached to it--to its feet, its knees, its arms, its head. It hangs by the strings. A puppeteer, hidden behind the little stage, holds a little control that pulls the strings. With skill, the puppeteer can make the puppet look as if it is alive. In the Walt Disney cartoon based on the book "Pinocchio," Pinocchio is a puppet.

So, figuratively, we can say someone is a "puppet," and the people secretly controlling him are "pulling his strings."

2) Suppose a group of people are eating at a restaurant. After the meal, the waiter comes and brings "the check"--the slip of paper showing what was ordered and how much the meal costs--and puts it down on the table. Someone has to pay. The person who is going to pay picks up the check from the table, and gives it to the waiter, along with payment.

"Pick up the check," then, means "volunteer to pay for the meal."

"Pick up the cost" then, is a figure of speech that can mean "volunteer to pay the costs."

One can imagine a conversation:

"Can we have a dance party in the gym?"

"I don't know. Last time, the kids left a real mess and we had to pay the janitors overtime to get it clean. I guess we'll let you, but you have to sign something saying that if cleaning is needed, you'll pick up the cost."

"barring" = to stop something to disallow it. It means Trump cancelled or was forced to cancel speaking except for on a few college campuses. Because his right wing advisers, made an issue out of some protests. 

Sometimes the word barring can be used similar to except maybe that is what is confusing. The use of two words so close together where one word has more than one meaning depending on the context.

The  violent protests were made by "non students" it means that it wasn't the students it was other people or organisation etc. Trumps right wing propaganda machine, complained about free speech problems. I do not know the actual event your text describes, but there were obviously violent clashes. Clashes are fights either between one set of protesters or the police and the protesters. 

The texts that you have problems understanding are full of the type of phrases Dan is talking about. 

Who made the violent protests, probably infiltrators, people who go to other peaceful protests to cause trouble and civil disorder. It is a global problem.