Bacterial bloom is a common occurrence in most starter aquariums, it typically clears up within a week. It looks "foggy" and kind of grayish. Once your beneficial bacteria establishes itself, it will quickly clear. You usually wake up one day and it is gone.
As for filters, the one you have can probably work if you baffle it properly, but if you find not you can get the Azoo Palm/Mignon filter on Amazon for about 20$. It's a great filter, but you'll want some different sponges and filter media for the long run, but the black one included is great for sticking out the exit for baffling.
As for your plants, the best bet is you get some Flourite or Eco Complete for substrate, but barring that I have grown swords in large gravel like you have with strong proper lighting, good aeration, and suitable fertilising via DIY Root tabs (Osmocote+ pellets in gel capsules) - easy to make yourself. One per 6 inches lasts 6 months or more. Be aware, you'll need to keep on top of water testing when you put these in for the first week incase anything spikes, be on hand with Prime.
You will definitely have to take them out from the sponges and spread their roots well when planting. It looks like you may have a couple plantlets of each, in which case you can have them in multiple spots. They are great, my bettas and bottom dwellers love hiding in them, and my snails love to perch on the leaves.
Another plant I've found to grow wonderfully in these conditions is Hygrophila, both Compacta and the Wisteria varieties. I have grown Bacopa Cardinalis, Anubias Nana, and floaters with similar situations with slightly less rapid growth. Oh, and Ambulia grows well, too.
Unless I am wrong, the right one is Java Ferns - they can just be tied to your rock display if you'd like. Sometimes I find they don't do as well 'planted' proper, just tied to things.
Plants are awesome and all of my bettas LOVE frolicking in them, so much better & they'll help remove nitrates from your water in the long run.
Good luck :)