Inspect the ABS unit for leaks of brake fluid, even very small wet spots, some the problems listed seem like pressure problems, the ABS pump is trying to keep the ABS unit pressurized but its leaking. The vehicles that correct itself after turning the engine off OR when the ABS light comes on, then the entire system is disabled and the pump stops, so no more feedback in the brake pedal. The leak can be an internal seal leak and you will not see it on the outside of the ABS module so beware. Locate the ABS pump and see how often it is running, sometimes it cannot be heard from inside the vehicle while driving down the road. Hook up a volt meter or test light that can be monitored, the pump should not be pumping too much.
On other vehicles when the VTC or other related systems have a problem, it will also affect the ABS and possibly the Emergency brake light, this does not mean you have 2 or 3 problems, only one system can affect all 3, Trac Control uses the ABS system to control traction so they are interlinked.
Many suggestions given are sound advice, dirt, metal fillings on the magnetic pick up, loose wheel bearings, too much air gap from the tone ring to the sensor due to improper installation or corrosion pushing the sensor out away from the tone ring where the spindle will need cleaned out with sand paper or other course cleaning procedures to allow the new sensor to be installed easily.
Under all circumstances, there should be a code in the computer but the problem here is, can the free code readers at Autozone or other places read ABS codes and TCS codes? I feel a paid diagnosis is better than swapping parts based on a guess. Specially if there is scan data showing the poor system operation, not always 100% but pretty accurate most of the time, an oscilloscope will prove out a bad part in the trained hands of an electronics technician.