Something that I failed to mention is to change the oil with the engine warmed up first and then drain it as soon as possible. This thins the oil so that it drains much better, and it also gets the crud in the oil in suspension so that it can flow out with the oil change.
Also, when running the gen it should be level. This is the same concept of filling it with oil. Splash-lubricated engines can be very picky about under or over oiling when the gen is off level.
Regarding the oil, I am not sure where you are seeing the non-synthetic oil being recommended over full synthetic. Honda does promote their oil, but then they (1) expect that the oil is changed on schedule, (2) they know the properties of their oil vs a cheap store brand, and (3) they get to sell you oil.
The manual states this:
So, an API SJ 4-stroke oil could be non-synthetic, semi-synthetic, or full synthetic. I will be less diplomatic than above comments and say to always use a full synthetic after break-in.
Years ago I was using non-synthetic oil in my lawn mower. At the end of the mowing season the oil would be coal black and nasty. I switched to full synthetic oil and now the oil at the end of the mowing season looks like it is new oil. I attribute this to the higher operating temperatures that full synthetic oil can achieve before breaking down. These engines are air-cooled and often run hotter than your car engine. I am not necessarily promoting Mobil1, but
here is good info on the differences between conventional and synthetic oil. If you don't mind the expense, there are also oils specially designed for small air-cooled engines.