Revs up when braking at sixty mph and higher and sometimes engine surges at idle

Try cleaning both sides of throttle body assembly

If you get a nice idle flare-up to 1500 rpm at engine start-up, the idle speed motor is working, and carbon isn't the issue. Consider watching the idle "steps" on a scanner when the problem is occurring. The Engine Computer places the AIS motor to one of 256 steps. As the number increases, the armature pulses to a new position, and the threaded shaft retracts the pintle valve to expose more of the air passage around the throttle blade. Step 32 is typical for a properly running engine. You'll find it near step 50 with a single cylinder misfiring on a V-8 engine.

What you're looking for is whether the step number goes higher or lower when engine speed increases. If the step number goes up when speed increases, the computer is requesting that higher speed in response to some other sensor reading, most commonly an incorrect reading from the coolant temperature sensor. If the step number goes down, even as far as "0", when speed increases, the computer is trying to reduce engine speed, but without success. If that is the case, the entire idle speed motor system is working correctly and you need to figure out why engine speed is going up. That is almost always due to a vacuum leak. Given your description of how the symptom acts, I'd be looking for a vacuum leak that opens up when the engine rocks and tugs on hoses.

Also don't overlook the need to relearn "minimum throttle" after the battery has been disconnected or run dead. That usually results in hard starting and stalling from low idle speed, but it can cause erratic idle speed too when the computer tries to overcome the low idle speed.

Yup. The Engine Computer controls idle speed on both throttle systems; the common sense, inexpensive, reliable cable and the ridiculously-over-complicated and frustrating electronic systems that put Toyota in the news a few years ago. It has to know when to do that, and it knows that by seeing the TPS voltage that corresponds to closed-throttle. That voltage must be learned because it will be different for every sensor that could be installed. Learning minimum throttle eliminates the need to adjust the TPS mechanically when it is replaced, as was the case many years ago when some aftermarket fuel injection conversion kits first showed up on the market.