Yes, your tank is cycled... if you use your current filter on the new set up, I think you won't have any problems.
The nitrogen cycle happens all the time in your tank, but what we call "cycling," which simply means building up the bacteria in the first place, takes about 6 weeks, if you start from scratch. But you already have an established filter. I don't think the temperature change will harm the bacteria either (as long as you don't take them out of cold water and plunge them into warm water). You can put the heater on the tank right now, so that when you get back, the water will be at or almost at the tropical temperature needed for tetras, cories, etc.
As for plants... it's a hit and miss. I'm terrible with them. There are some hardier species, but unless you know what you're doing, they're probably gonna die. If you want to, though, you can go ahead and try. The best substrate for plants is potting soil, with a layer of gravel on top.
In your tank, you could have six tetras of one species, six of another, the beta, and maybe 4 cories or so, the kind that stay under 2". If you do 30-40% weekly water changes, you could have a really nice, well-maintained set up.