How to increase participation in tutorials, as a teaching assistant?
I often walk over to a randomly picked student and ask him/her a question. I do that randomly and unpredictably, including corners of the room or sides where students expect to be out of the action, and no student will stay inattentive. Under no circumstances belittle or ridicule, though, no matter how wrong they are, and if the student makes even a partial step in the right direction, give them credit for this. If they are stuck, you can then solicit help from others to further complete the answer.

For students avoiding your eyes, try to pick a question with unclear/ambiguous response. When you then ask the student, they are likely to respond with "I don't know" or similar, and you can then say that they are indeed right, that the answer is indeed unclear (and, of course, you discuss why). This gives them a confidence boost, and the next time you ask them they will be less worried to answer.