Smaller tanks take much longer to cycle. I have two 5 gallons that used to have a betta in each one.
Honestly with the plants and only stocking a beta I would do a small water change and wait a couple days till your nitrItes are at zero and then put your beta in. You probably have plenty of bacteria along with the plants to handle his bioload. I always just popped my betas straight in with plants and just did water changes if anything got out of wack, no different than changing the water in his bowl? I'm a fish in cycle type a girl though
As for the filter, no the betas don't like it at first. Most are used to living in a jar so you gotta give them time to get used to it. Once they are they love it. I had one woth an internal fluval minifilter and I just turned the flow on the lowest setting. Another I had a small hob
filter and I fixed a sponge on the filter outtake that distributed the flow so there wasn't much current.