It is stressful for the fish and I'm a big advocate of the 'fishless cycle', but I don't see it as being cringe worthy to cycle it with fish. Many people do just that but without plants and have many deaths. I also think the home you have is FAR superior to a super market chain. So even if you do cycle it with the fish in there, I think it is better than some of the other alternatives.
Where are the shots of the corys?! They are my favorite little guys. Also, if you need some floating plants, send me PM. I'll give you some frogbit to start off some of your plant mass.
Absolutely. If you are doing regular water changes and keeping levels down, it might not be too bad. I've seen people run without filters and do only water changes, so you are inline with that. Any plant choices beside the moss?