The collar is the metal ring on the actuator that "chokes" the rod that you use to move the actuator from above (btw, I looked that part up - 163786, it's not available anymore, so if we find something wrong with it, PM me and I can ship the one I have here.) so they both move as one.
It should move up noticeably. I'd say close to an 1/8 in. If you put your finger on the top of the plate - what you would see when the machine is standing in it's normal position, when you're moving the actuator manually you should feel it move. It's hard to be in 2 places at once, I know
You won't actually see the bottom portion move, that's the gear for the bobbin case. The part that moves in and out is pushing in to the plate above that gear. Hence the finger
If that piece under the bobbin case doesn't move, it's a different problem than if the lever's not doing what it's supposed to, so we need to be sure.
The screw you found on the right is called a set screw. You can't find the head because it doesn't really have one. It's got a recessed area, where a hex key fits. You may not have a set of these if you don't do repairs to a lot of different sorts of things.
Without being able to check its tightness, you would have to watch the shaft to see if it's moving at a different rate (called "slipping") than the actuator, that's where the "D" shape is really handy, but you can also draw a line with a marker or something and watch that move (or not move)
Correct. The actuator is what I gave you the original part number and link for - plastic, triangular with a metal collar...
Inside the actuator's metal collar, you can see the rod that goes to the top with the cream / red on it. On mine, the rod has a D shape, yours looks to me like it's actually round, but I want to be sure.