I write everything on paper, although I duplicate all the important parts to LaTeX. This means that I go through a notebook approximately every 3 weeks, but it's definitely worth it. I only use LaTeX to communicate with others and store really important bits (although recently I've also just been scanning my paper). I believe this is pretty much standard among my peers.

I've done different things at different parts of my life. I used to work everything out on paper first, but that got too time intensive. For most of grad school I worked in LaTeX exclusively, but I gather from my peer group that being able to do this is kind of rare. I bought a Surface Pro 2 a couple months ago, so now I do both simultaneously, sketching out bits in windows notebook while writing exposition in LaTeX.

I don't think there's really a general best practice to be found here; I think you just need to try different things until you find a workflow that you can live with.