It's been a LONG time since I worked on a 380, but I'll take a stab at answering as many of your questions as I can.
1) The 4 position switch is your light switch. One position for headlights, one for rear lights, one for both front and rear and one for ...something else that I can't remember. :ashamed:
2) I don't know what the pull out switch is for. They didn't come with a pull out switch
that I remember. PO could have put that in for the headlights at some point.
3) That button is for the horn or on some units brake lights. Yes, some had brake lights. The 380CK is really an 880 David Brown that's been rebadged and painted Case yellow. The DB tractors were built in England and had some things on them that we didn't use here in the states.
4) You are correct. The 3 position switch is for turn signals.
5) That lever is the fuel cutoff as you've already found. It should always be either completely in or completely out. It doesn't work like a choke. It needs to be either on or off.
6) That lever is to engage the differential lock. Engaging the diff lock will make both rear wheels turn at the same time. Never engage the diff lock with one wheel spinning and the other stopped. If you need the extra traction, stop both wheels, step on the diff lock lever and then slowly let out on the clutch until you feel both wheels engage, then you're set. There's no need to stop when you take your heel off the diff lock. It should be spring loaded so that when there's no longer any load on the rear wheel, it unlocks itself.
7) PTO engagement lever should be behind you and on your left if you're sitting in the seat. It sticks up from a shaft that goes into the left side of the PTO housing.
See attached diagram:
8) It has A glow plug, sort of....

It has "cold start assist" which is a single heater screwed into the intake manifold that heats air and fuel inside the intake.
You can go to the
Case Construction website, click the "Launch Parts Catalog" link, then type in 380CK with no spaces to view a complete parts catalog of your machine.
Hope this helps!
Mark