I had a Honda 500 watt generator I got in the early 80's. It could have been a 600, but my mind says 500. Not important, essentially the same little generator.
Powering the Univolt was possible, but the Univolt is such a poor charging unit and with it's ferro resonant transformer the little Honda did not like to fully saturate the windings on the big transformer it contains. It would never put much charge into the batteries. So, what I did was to use the Honda 12 volt
battery charging outlet directly. I believe that it is an 8 amp output. It used a special bent prong plug for the output, but with a pair of pliers you can twist any regular two prong plug to fit the outlet. Then make up a cord and watching the polarity, connect it to your AS battery. Put a fuse near the battery for safety.
On question #3, I am not sure. I now have a PD 9245 in each of my rigs, and my current Honda EU 1000 inverter generator will operate them (but not a PD 9260). I don't know if that little old Honda will be happy with the PD converter/chargers or not. Mine got sold with one of my trailers, so I no longer have it to test with.
In theory all 120 volt 60 Hz power should be the same, but in practice the little generators do not "look" the same to some loads like the old Univolt and even a larger new 3 stage converter/charger as they do to a big power company system. Complex Electrical Engineering reasons that I only have a slight grasp on.
But, hook the little Honda up with the 12 volt
charger directly and you will be fine. It will try to stuff 8 amps into the batteries as long as you keep gas in it. In fact, you can overcharge a battery which is not very run down, so don't overdo it.