I don't know why this is but somehow, whether on keyboard or on gamepad (with joy2key) when I press the draw button, Lara will jump at the same time, sometimes even discarding the draw command, which is especially gratting when you're trying to get your weapons out and she just jumps around. Any idea why this is?

I had an issue like this in games 1-3. I liked to change my jump key to space, and whenever I loaded up a game it would have Lara draw her weapons as well as jump. To fix this, I would just go into the key binding options and spin it to "user keys". I'm not sure if it will help with a controller as I've never used one, but it should do the trick for a keyboard.