I use a pen and paper at least once a week when I'm at university but as I'm on a computing-based course I use computers a lot more than a pen and paper. When I'm at home I almost never use a pen and paper, I used to occasionally use them for taking down notes but now I usually have a laptop near me which I find it easier to make and store notes on.
No, not even when I was in school. My handwriting was so illegible, that I convinced basically every teacher past grade 3 to let me type everything. Personally, I like it better anyway, it's faster, cramps hands less, and looks more professional.
I like how good handwriting can look, but I'll never do it myself. Even little notes I use a text editor or stickies on my computer.
I still like to lay down or sit down in a quiet place and just right random thoughts. I'm also a huge fan of making lists and it's easier to do those (and more fun) with pen and paper. I always carry a small pad and pen to write notes of random things to check out or remember. I def. agree that sometimes the thoughts flow a bit better when writing them out then when sitting in front of the computer.