Sorry for the late reply to your posts - my iPad keyboard died on me.
First, the cycling. Cycling basically means maturing the filter so that lots of beneficial bacteria grows. This beneficial bacteria eats the toxic ammonia that fish excrete and turns it into slightly less toxic nitrite. Another type of beneficial bacteria also grows and that eats the toxic nitrite and turns it into even less toxic nitrate. Without a filter and hte filter media, you really can't cycle a tank because the beneficial bacteria has nowhere to grow.
To get your tank to cycle completely, it would be best to get a filter rated for 10-20 gallons. That would provide enough surface area that plenty of bacteria can grow, enough bacteria to handle all the waste in a 10g. But even with a smaller filter, you'll still get quite a bit of good bacteria. :)
It's possible. I don't know if I've ever seen one of my fish sleep-swim but I wouldn't really put it past them. Since they don't have eyelids, it's very hard to know exactly what they are doing when they are asleep. It could also be that since he was being fed later than usual, he had to kind of "wake up" first. The same way we humans can be really sleepy still for a few minutes after we wake up.