I applaud your efforts to cycle your smaller tank. This is ultimately best for your fish, in my opinion.
Your cycle has crashed twice, it seems. Not unusual when dealing with low ammonia readings. What are you using as an ammonia source?
A few tips to speed things along:
---keep your ammonia >2.0ppm until you get nitrite and nitrate, then back off to <1.0ppm
---get your temperature up around 82*
---run your filter on high to increase aeration
Your tank isn't really cycled until you're producing nitrate, with ammonia and nitrite at 0.0ppm.
The hornwort is your only plant that eats enough ammonia to mean anything. Just keep it. In fact, getting more fast-growing stem plants like Wisteria, and floaters like watersprite or frogbit will enhance the safety of your tank once it's cycled, by preventing ammonia spikes and mini-cycles. Don't know if that dwarf lily is a fully-aquatic plant. Check the plant thread for more advice.
I forget what the little white dots are, but they're not dangerous.
Keep us apprised of your progress.