I've always wondered this, if your head coach runs a 3-4 defense can you still run a 4-3? Or are you required to run whatever your coach's preference is?

Well you are forced to use the plays that he has available, and beyond that I would guess it depends on how much you wish to manually edit your defensive gameplan. If you look at the plays they will tell you if the defensive play has 4 down linemen or not, you can select those plays but I think 3-4 based coaches will tend to lean towards 3-4 plays (not totally sure on this) if they are selected in your defensive plays.

You can run whatever is in his playbook. 3-4, 4-3, 46, Nickel, Dime, 3-3-5, whatever is available in his playbook. The point of different coaching styles is that not all plays are available to every coach.