If your notes are so messy that they require rewriting, then you need to work on a better system of note taking. I have never rewritten my notes, and they are always clear and organized. Time is definitely much better spent studying or practicing problems versus rewriting all of your notes.
IDK about this, actually. My notes in class are usually horrendous, but it actually really helps me to go back the next day and organize everything into more coherent notes since it means thinking back to the lecture and getting everything into long-term memory.
OT: Yes, it is absolutely essential that you take notes in class, unless you have truly superhuman memory and can remember everything your professor says. The process of taking organized notes enforces active and engaged listening. You also need to follow along with demonstration examples rather than just watch them being solved.
Plus, it's pretty much the universal signal to your professor that you're actually paying attention, which is kind of important if you want to stay in their good books.