Responding to Ultradog's comments:
I think I can see that it does have diff lock. In the pic taken from the rear, just to the right of the mid section, on top of the axle housing you can see something. That should be the diff lock linkage which goes down through a hole in the axle housing to do it's job.
On the tranny I'll say it's an 8 speed. I have a 6 and my levers are reversed; tall and short levers are on opposite sides. Short lever is hi-lo range and long one is gear selector.
On live vs tranny PTO I'm going to say it's live. Reason is the height of the clutch pedal. Mine is live and to get the functions lined up the pedal has to be uncomfortably (somewhat but I am used to it now) high.
FYI, on this tractor you only have two fluid compartments with the full measurements made when fluid comes out the ports UD circled in red. The fill ports are easily visible up on top of the tranny and top right rear of the mid section at the rear axle.
With it having a dry air cleaner it is a later model. Mine is a '65 and is oil bath.
To check if the "proofmeter" (tach) is working, to the right of the numbers is either a black or white bar. While the engine is running, this bar will rotate and change colors which is your clue that it is functioning.
It's cold now and a starting aid may be required but hard starting under mild temperature conditions is an indication of worn rings. Mine and surely yours has a starting aid which is a hot wire and fuel release valve located just behind and inboard of where the air inlet hose attaches to the intake manifold. When activated, key position or pushbutton (depends on tractor year model) it heats a wire coil cherry red which causes a valve to open and dribble some diesel on that hot wire which catches on fire. The cranking engine sucks these fireball drops into the combustion chamber and lights off the engine.....literally.
The 3000 is a fine utility tractor and I wouldn't swap mine for anything.....for the intended purpose.
On the "Live" PTO. Depressing the clutch all the way stops everything. If the PTO lever is pulled forward when the clutch is let out about half way the PTO will start spinning but no ground speed. Letting the clutch on out the tractor will move across the ground. For doing post holes, you can't beat Live PTO.