Firstly

to our happy home here.
Yes you are right they are not a common mower.I don't believe there were many produced to start with.
One thing it may well be, Did you put back the original jet back into the carby? As these guys were only 85cc so they only required a smaller jet.Putting a jet that would run say a 160cc engine could cause this problem.After all it will get fuel to start and then suffocate due to too much fuel.Your question (the tube from the jet screw (white) is in the jet screw itself is that right?)Yes for the G4 carby they do have a white stem on them.Yes this mower is suppose to have a G4,but as I said before the actual hole size is different.So if you haven't changed it from when it was running,you should be good.
Also the primer bulb should be intact,they should only have a small hole in the middle.I had one not long ago that had a tear right through it.The mower did run,but not well,also dribbled a lot of fuel out of it.
Your question Could the needle be sticking closed,and is the orange one I have right? It used to work.Yes the needle could stick especially if it is dirty try and clean it out.I can't remember now what colour needle they had in them.If it had the orange one in it that's the one I would put back in it.I would have to go and look at mine to see what colour they were.I could do that soon if you get to stuck.
Your question and Is my black poppet valve in the carb right assume there is no 'A,B,C' adjustment that is possible with this poppet valve? Not real sure what you mean by poppet valve. I'm thinking your talking about the diaphragm at the back and I don't know of any adjustments there.
I'd also be checking the plastic hoses as well if mice have chewed other parts they could have chewed them.Like the hose from the carby to the decompression valve.Hope this helps a little.