Do vector drawings colored in with Illustrator’s Live Paint work as expected for use in a responsive website?
Yes, but you probably need to Expand the objects in order for proper compatibility.
You can use Live Paint to color everything in.
Now go to Object → Live Paint → Expand in order to expand the paths into regular paths that are independent of the Live Ppaint group.
(You can also use Object → Expand and only select Objects – it seems to produce the same results)