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I am a hobbyist graphic designer, looking to recreate part of this image, specifically the top half with the sun: 
Right now, I made a crude copy using line segments, and filled them in with live paint bucket. However, I liked the look of the waves at the top, and was wondering how to get those with my image. So if anyone has any ways to distort the lines without ruining the circle, that would help a bunch.
Thanks!
Once doesn't "distort lines without ruining the circle".
You draw the lines then clip them to the circle.
This is merely a collection of single paths.
The Shape Builder Tool is used to remove portions, by holding down Option/Alt while dragging.
Then the Live Paint Bucket Tool is used to fill the remaining shapes.
I'd take a different approach. Live Paint is fun, but it has some limitations.
Instead, I'd draw the lines with the Curvature Tool, clean up and make the individual pieces with the Shape Builder Tool.
Example
Then fill each piece with the desired colour
To answer the question:
Live Paint allows editing the shape borders and the paint follows as long as the paint is not expanded. After expanding you must edit the shapes separately.
In the left there's a non-expanded Live Paint result
In the middle a couple of new anchors are inserted to a borderline and the anchors are dragged a little off with the direct selection tool
In the right the anchors are made smooth with the anchor point conversion tool.
BTW Check from the Illustrator documentation (by Adobe) how to draw and edit curved paths. I guess most of us would draw curved paths from the start for this purpose.
