Ok, I'll try to answer some of your questions.
Yes, the drive shaft on a hydro tractor spins all the time, so yes, that little fan under the tunnel cover should spin any time the engine crankshaft is turning. They are actually coupled together with flexible couplers.
The little Kubota diesel engine is a FABULOUS engine. They are nearly bullet proof, and have TONS of power for a garden tractor. The starter is a Hitatchi, and they do sound different. Make certain that you have a really good battery with a strong charge. Do not try to start with a weak battery, you will burn up the starter. If the starter sounds really bad, pull it and have it rebuilt. The last one I did was about $50 to be rebuilt.
Because it is a diesel, there are glow plugs. The engine will NEVER start if the glow plugs are not used to heat the fuel prior to starting. The Kubota is all cast iron and way too cold natured to start without glow plugs. I believe (but not sure) that you turn and hold the key backwards, opposite of the start/run direction, to activate the glow plugs. Also, there should be a chrome thumb hole on the dash, where you hold your thumb while holding the glow plugs on. When your thumb gets hot, the glow plugs have done their job.Then you can try to start it. It should look like this:
If no thumb hole is present, there may be an indicator light instead, which should brighten as the glow plugs get hotter. Don't hold it too long, as you don't want to burn out the glow plugs. Read the manual first!
Your tractor appears to be the Cub Cadet 1772 diesel. Be careful, they made that same tractor with a Kubota, 3-cylinder GASoline engine. The differences should be obvious, but believe it or not, it's really hard to tell the difference between the two because the fuel injectors on the diesel look a whole lot like the spark plugs on the gas engine. Just double check!!!
Anyway, go here and download a owners manual: http://ccmanuals.info/pdf/1772%20Owners%20Manual.pdf
and service manual: http://www.slideshare.net/alicediana/cu ... -1606-1806
If it's sat for a long time, you might as well start by draining the fuel tank and getting a new fuel filter to install. Remember, diesel's are not like gas engines, you'll need to prime the fuel filter and pump, as they typically do not self-prime like a gas engine. It's a pain in the a$$ but you don't have to do it very often. A diesel engine will NEVER start if any water is present in the fuel. They only start on compression, and water can be real bad in there!
Take your time with it. They are WELL worth the time to restore!