What causes traction control to keep sticking and brakes go low when brakes are hit hard and the car makes a noise and shaking like tire rods are broke?

Did it stay on or just flash a second? Anything the interferes with correct wheel rotation can trip this warning. Did you change tires, have brakes serviced or hit a pothole real hard? Believe it or not even a weak battery can cause warnings of this type due to voltage fluctuations. There are sensors in you wheels that monitor rotation speed, braking and position. It sends this information to the cars computer so that it can control speed of engine and braking to prevent skids , wheel hop, and help keep the vehicle under control.

Can you grab the tire/wheel assembly and it has looseness while stationary or is it a shake that starts when speed increases?
If it starts to shake as speed increases,you may need a wheel balance.
If there is any looseness when you grab it while stationary,get a look under there SAFELY!! and replace any loose parts.

• Knowing the warning signs of impending trouble can go a long way toward keeping your car from a highway breakdown.
  
  You need to develop an awareness of how your car should sound. For instance, when you turn the key to start, the gear like sound of the starter motor engaging should be smooth and sound like an electric motor working hard to crank the engine. If you notice a different sound - more like a grinding - chances are you have a starter motor problem about to happen. Let's say you're driving along and you feel the steering wheel vibrating at certain speeds. The onset of this subtle vibration could tip you off to a tire that needs to be balanced, worn steering linkage or a bulge in a tire that may be about to blow out. In any case, the vibration needs attention. Normally you shouldn't notice the sound of the exhaust system. A sudden change in your car's sound, therefore, isn't something to ignore.
  
  Suppose you're used to a pleasant hum, then, perhaps after a bump or when starting up the car one morning, there's a loud roaring that sound like a hot rod without a muffler. The harder you press the gas pedal. The louder the noise becomes. A noisy exhaust means the exhaust gases are blowing out under the car rather than at the rear of the car - a dangerous condition. Brakes play their own warning tune. On many cars, a scratching noise coming from the wheels that stops when you stop pressing on the brake pedal is caused by a low-brake warning device. Its purpose is to warn you that you need front brake pads. A loud grinding noise when you step on the brake pedal is an alarm that your brakes need immediate attention. The grinding noise occurs when the brake pads are totally worn away and the metal of the brake pad backing plate rubs directly on the brake rotor.
  
  Ignoring grinding brakes can be a costly - or even deadly - error. Shock absorbers help keep the car steady and balanced while the wheels roll over bumps and ruts. Their modern counterpart, the MacPherson strut, does essentially the same thing. You might not notice worn shocks or struts when you're simply riding around town. The safety effect of these devices comes into play at higher speeds. When you hit a bump, the tire wants to bounce like a basketball. Worn shocks or struts allow the ties to bounce out of control making steering difficult and braking less effective.
  
  What you may notice is the car rocking more every time you stop, assuming an almost boat like quality. Old, broken or worn out shocks and struts should be replaced to ensure safe steering and braking at higher speeds. Engine noises are easiest to hear when the car is not moving. Transmission sounds or brake noises usually occur when the car is moving. Pay attention to any new sound and keep notes - how it sounds, when it makes the sound and what difference, if any, occurs in driving when the noise is present - to help you describe it to your service dealer.

yes. if the axle is bad and binding it can cause the front end to vibrate because it gets out of balance.

you should keep all four tires on the ground !!!! LOL ! your rotors or brake pads are bad. do you feel it shake in the steering wheel ? Check these three items and im sure it will fix.......1) jack front car up and try to move front tires in all directions, does it feel loose at all? if yes check tie rod ends, upper and lower ball joints. 2) check brake pads for wear, heat from pads warp the rotors and cause the break pedal and steering to shake or vibrate. 3) If nothing can be found wrong with the tie rods or ball joints i recommend changing the rotors and brake pads, a pretty cheap fix.

tie rod more than likely attached to the rack and pinion

could be a bd axle or ball joints wheel bearings are also a concideration

you going to need a scan tool to check for the problem. a sensor has gone bad

Bad shocks will cause the front jump around. Also check the tires for uneven wear.
The entire front end should be looked over if the shocks are bad you may also have other parts worn as well.

if you hit the brakes and it shakes its your rotors, they're probably warped. Having them turned (polished) won't fix it. You'll have to replace them. As for the shaking at over 50, if you're not touching the brakes, you'll probably just need to have your tires balanced. Maybe jack up you car an check the ball joints and tie rod ends (keep the tire on and see if there is play, either up and down (ball joints) and left and right (tie rods)). Hope this helps.