The Grit Fluxx comes with a non-integrated headset (it looks like some might even be threaded, which I would argue is not HIC). Integrated headsets should be interchangeable, but I don't know if non-integrated headsets are. They might work, but you should try it out before buying. It's definitely not integrated though.
It sounds like you needed those spacers underneath your headset's dustcap to account for an improperly machined bearing cup. This is fairly common. The top of your headtube should go like this, going up: bearing cup (integrated, hopefully), bearing, headset spacers, dustcap, then bars/compression/clamp.
You can clean it with soap and water, but you'd need to dry and regrease all the bearings (headset and wheels) afterward to ensure that they don't rust. Better yet, just use a wet rag to clean off your deck being careful to not get water in the headset or wheel bearings.
It sounds like your compression isn't tight or you might be missing some parts. Make sure that everything is properly seated and installed, then tighten your compression so that it is snug but still turns freely.
Yes, you can install a threadless fork, but you'll also have to replace the headset with a threadless headset and use some kind of compression. Depending on what compression you go with will determine what kind of clamp and bars you can use. So, yes, you can do it, but it's a lot more involved than just replacing the fork.