Can I draw my own assets and use it here?
Kinda! Not by directly replacing anything built-in, but by using the Customization Menu. You can import images. You can then make the images jiggly, you can pin to the model, you can hide the object you're pinning to to "replace" it. It has lots of options.
But if you're looking for complete customization of the entire model, vTuber Kit cannot do that.
Is there a way to make it so that the custom items don't sling around? For example, I wanted to put the halo kind of around my character's head at a diagonal angle. But despite turning jiggling off and pinning the item to the hair/head, whenever I shake my head the halo is slung out of place. Is there any way to make them fully attached to the head to where they stay in place and move at the exact same time as the head, rather than acting like a rubber band whenever I move? I hope this makes sense!