I had my 5 gallon tank up and running for a few weeks with plants in it before I decided to put my betta in it (I didn't officially start cycling it yet). Within 3 weeks, my tank was fully cycled and I think it was because I started running it with plants in it. Anything living will get bacteria started.
Anyway, live plants are difficult to grow if you don't have the right lighting. I never use fertilizers or CO2, but lighting is really the key. I was able to grow some plants when I had "low light" (which was a 60 watt CFL bulb with a 2700 K rating. you need to check the bulbs for K rating). the higher numbers are whiter light, which is closer to daylight. You need at least 6500K for growing medium to high light plants. I ended up with a really great LED desk lamp that really caused my plants to start growing. But in my low light situation, I had an amazon sword, a rosette sword, hornwort, and fanwort. They all grew slowly and survived, but many other plants I had tried before that did not.
But in any type of lighting, I recommend fanwort as a floating plant. I just really like how nice it looks. Also, Rosette Sword and Amazon Sword. Supposedly Water Wisteria is supposed to be super easy to grow and supposedly grows like a weed no matter what, but almost all of mine has died over the past month since I got it. It's really fustrating.
People say java fern and anubias are good low light plants, too but I don't have any personal experience with those.