Salvinia, Water Lettuce, Amazon Frogbit and Duckweed are true floating plants, but you can also float Anacharis, Water Wisteria, Water Sprite and Hornwort. There are other plants you can float but these are the ones off the top of my head. As for rooted in the substrate you could go with Hygrophila Polysperma/Dwarf Hygro, Crypt and Vallisneria. You could probably put a small Anubias and/or Java Fern in the substrate just plant the roots not the Rhizome.
Some of these plants might grow to tall for your tank though.
You Have to tie them to something. Eventually, their roots may grow around what they are tied to. Rocks, driftwood, the log you have in there, anything. You can use fishing line or thread. I usually use cotton poly thread that is the same color as what you are tying it to.
hmm I don't have any string or fishing line lying around; plus, I really like their placement. I may just leave them there and see how they fair. What signs should I look for that the plant isn't "happy?"