If a betta doesn't make a nest it can still be a happy betta. our 2 males haven't been doing any bubble nesting in their 10g tanks. sometimes a bubble nest is him in the mood, a storm might be approaching or he's just a betta that likes to build bubble nests. it's fine for him to do as much as he wants.
if you want to speed up the cycle (and lighten your wallet), get biospira and cycle the tank fast. Otherwise you could do a fishless cycling on the tank.
One way to do the move is: When you move the betta, scoop him into a big bowl/cup and
acclimate him to his new water. empty some of the water in the bowl carefully so he doesn't fall out, then float it in the tank, adding some tank water to the bowl every half hour or so (you can vary the time somewhat i've found). after 3 or 4 times adding tank water, you can let him out if the water temps are nearly the same, which they should be by then.. keep the tank lid down when not adding water so he doesn't jump out of the tank.