I don't know where you got the idea that Kubota has had issues with the 12 x 12 gearboxes. The 12 x 12 has been very trouble-free and is the most popular option.
Not necessarily; as the engine warms up, the intake can seal due to expansion.
It's rare, but it happens.
The first check should be to visually see that the choke butterfly is fully blocking the intake, though.
Remove the 4WD detent retaining bolt shown below, see if it has a spring and ball under it.
I suspect it doesn't.
The ball goes in first, then the spring.
They also do not have an electrical shut-off solenoid, just a pull cable.
Did you by any chance kick the left rear tire? It works more often than one thinks!! ;)