Well, I hate to break it to ya, but those plants aren't aquatic, at least not fully aquatic. They may survive for a little bit, but they'll end up rotting and making a mess of your water. There are some people here selling plants if you'd like to look around the marketplace. Some plants I suggest for your set up: Dwarf Hairgrass, water wisteria, Amazon Sword 'Tropica', java fern, anubias, cryptocoryne, just to name a few.
If you're keeping with window light only, be aware that you're going to get some algae and that most plants need more light than just a couple hours, six hours is minimum for low light plants

You can definitely get away with Java Fern and Anubias in there with just window light, but again, algae because of the soil you have in there, it's seeping excess nutrients which slow growing plants like java fern and anubias won't suck up so quickly. It's not the end of the world though! You can also add floating plants like Dwarf Water Lettuce, Frog-Bit, or Salvinia to aide in keeping algae at bay! :-D I think it'd look super cute if you had a nice big bush of Java Fern in there with one nice big node of Dwarf Water lettuce on the side.
All that aside, Betta's are tropical fish and they do need at least heaters. Filters aren't necessary (though they do help out the fish keeper), the heaters are, unless you live on or near the equator. There are some small cold water fish you can keep, but most of them need more than just one gallon unfortunately. Or you could do Shrimp! They need a very well established tank though, especially if it's small; you need it up at least 3 months or so and growing well before adding shrimp--if you want to do it successfully that is, you can add them early and you might get lucky, you might not, it's a gamble sometimes.