Q: Hard braking causes steering wheel to shake

It seems that you may be correct. I am here to help and hopefully find your issue. Find a road where you can get up to around 30-40 miles per hour and lightly press on the brakes. Press on them enough to where you will stop within 100 yards, so not too hard. Hold steady pressure and see if at some point you feel the brakes let off and come back again. If this happens the you will know you have warped rotors. I would suggest replacing your pads and rotors at this point. However, don’t stop there. Let’s look under your hood. Is your brake fluid low? If so, fill it. Let’s look at your power steering fluid and fill it up if it is low. Make sure you use the correct fluid. You probably want to jack your car up for these next steps. Let’s look at your steering components, do you see any torn boots on your upper or lower balljoints? Look at your tie rod ends and see if they are dry or torn, and then we want to look at the rotors now. Do you see any excessive wear on your rotors? Run your finger nail across them, did your finger stop at some point or do they look excessively rough? If so, I would again replace the rotors and pads on the front. I hope these few things will help you find your problem and get your car back to where you feel a little more safe again. If you need help replacing the rotors, I recommend having a certified mechanic from YourMechanic come to your home or office.