You're supposed to replace it when you rebuild the case - it does require a special tool or some creativity to remove and install it properly though.
The main-shaft there's really not way to check it's condition outside of twisted splines, pitting or scaring damage of the journal surface and making sure the oil passages are clear of debris and free flowing.
The input shaft, you want to make sure there's no major scaring/gashes on the sealing surface, no chipped teeth and you'll want to replace the pocket bearing.
Pump wise, I haven't the slightest clue; I've never ran into issues with one outside of the screen getting clogged or pump rub. They're a gerotor pump and fairly reliable; however, I'm not aware of the specifications to check one - sorry not much help on that note.
I will say one big issue you has is oil flow though. Oil shouldn't cake on the input shaft like that. I improperly seated a pocket bearing in my input shaft when I was testing my custom 246 and it pumped all the oil (2.3Q) into the transmission in under 50 miles - mind you it was still seated, just cockeyed slightly; it still had the pressure and flow to push oil in a very small gap.
So it is possible that pump is bad, the screen was really clogged or the o-ring into the pump was bad. You likely can submerge the screen in atf then manually turn the oil pump and see if it squirts oil - I just don't know if you turning it by hand would prime it though, I have never tried that. That's my ideas at least, maybe someone can chime in on that note.
I think what more than likely happened is the fork pad got worn down and then ground into a powder> then started clogging the screen> that reduced oil flow to the syncho> then that got grounded down and really blocked flow> then it started getting hot and cooking oil on the input shaft. That's my speculation at least.