Yup, the plants and wood are fine. Might want to keep an eye on the plastic curly tall one, whose name starts with a 'V' and I can never remember. Plastic tends to tear at betta tails, not always, but definately has a tendancy.
You probably can remove the aerator entirely. Looks like you've got a couple non-anabatoid fish in there (can't tell who they are), but the betta will tend to breathe from above. I didn't have an areator in my 10 when I had ghosts and cories in mine, along with my betta, for several months. Aerators increase the surface area of your tank, thereby increasing the transfer of oxygen into the water, and there is minor transference of oxygen as it passes from the aeator to the surface... but that's about it. I finally gave in and put one into my 20g, simply because I added a sponge filter to help with filtration.