I have these super weird yellow-green balls in my soil. Zoom in to see the detail. At first, it was a cluster here and there on top of the soil. After digging deeper, I discovered it the tiny green ALL OVER the sides and in the dirt.

It's a Dragon Tree in high quality soil (with organic matter). I use a nice fertilizer. I water once a week and the plant looks healthy.

HELP! What is it??? What should I do???

Its a fungus that is busy breaking down the organic matter in your soil. You should water only when the top 2 inches or so of soil feels dry.

How should I get rid of the current fungus? I've removed the majority of the soil, but there still seems to be fungus trapped in between the roots. Should I use a special solution?

Also, do I need to quarantine the plant so that it doesn't pass it along to my other ones?

Probably nothing to worry about. Just about all soil in house plants as well as in the garden has lots of fungi and other micro-organisms in it. That's what healthy soil is all about. Once it's used up all the stuff it's feeding on, it will disappear magically.