Shane, chill out. Stop and take a breather. Bugs are normal. All tanks have them. Mine has them, Cynthia's has them, everyone has bugs. They just show up. It's completely normal and HEALTHY to have bugs. If you go look at a pond, the water's not crystal clear, its full of bugs and algae and lots of things.
If you didn't clean the tank totally, your cycle should be fine. Did you replace the filter media? If the cloudiness has gone away, it could have just been stirred up dirt, airbubbles in the water, or just general cloudiness.
The 'mickey mouse faced' brown ones are probably planaria. These thrive on waste in the tank, uneaten food or axie poo. Make sure you vacuum the substrate on a regular basis.
Your axolotl can live in a bucket indefinately, as long as you change most of the water on a daily basis. It'll be fine. Indiana Axolotl colony are all housed in fish bowls.
Don't chuck your plants. Plants are good. We like plants. Plants don't 'cause' bugs.