I have a rough idling problem and when I apply the brakes, it shakes. What's up?

Diagnosing the rough idle when warm from an armchair is hard. I can tell you that you have (at least) 2 seperate problems. Rough idling when hot is engine related, and grinding/shaking in the body during braking are not connected. Where does it shake when you apply the brakes? Front end wobble or like the whole vehicle vibrates? To figure out if it is front or rear brakes, apply the brake pedal at about 45 mph and see if you notice the steering wheel shake/move side to side. If it is front end wobble, you should have the rotors checked for runout, the wheel bearings repacked and the steering and front suspension checked for slop/play. If the wheel does not shake and or move, but you feel it in the "seat of your pants", then try applying the parking brake (carefully) at 40-45 mph without touching the brake pedal. To do this, keep the park brake release handle pulled out and push reasonably hard on the park brake pedal with your foot to apply them. If it is in the rear brakes, the vibration will be there without the front brakes applied. If it's rear, have the drums checked for out-of-round problems, and the rear shoes and hardware checked for wear and or loose/missing hardware. May also have a rear differential issue.