1.... Why didnt the pilots put the plane into manual mode and land it at the nearest airport after they started having significant problems?
Well, there's no "manual mode". The instructions for runaway elevator trim is to turn off the computer-controlled elevator trim, but that also
shuts off the trim buttons on the control column. Then, you have to trim the trim wheel by hand. If you are already fighting extreme control forces due to mis-trimming of the elevator/stabilizer, you may not be able to take your hands off the column -- you need all 4 arms to counteract the control forces.
The problem is that the old software (not revealed to the pilots in the flight manual) wound continue to apply more nose-down trim every nine seconds, leading to ever increasing control forces, unless the pilot immediately applies a LONG countering nose-up trim. The crash pilots didn't counter the nose-down trim with a long-enough nose-up trim. So, the control forces became heavier and heavier with each repeat of the sequence.
2...What exactly was the fault with the angle of attack sensor?
These are easily damaged when moving air bridges and trucks around the aircraft. Apparently, the replacement sensor was improperly installed at the wrong angle, even though the bolt holes are designed to prevent this. One sensor apparently indictaed something like 40 degrees nose-up as soon as the aircraft started moving. Due to the airlines not wanting to permit ANY additional training for pilots moving to the Max, the flight control computers were set up to use ONLY ONE sensor at a time, depending on which FCC was the master. So, even though the aircraft had two sensors, it only looked at one or the other. There was no "alpha disagree" warning, even though the info was available to the computers. The reason was "that would require a change to the flight manual" which the airlines did not want.
3...Regarding software failure systems in passenger aircraft...how many other ones are unknown to pilots other than EMACs used in 737 MAX?...ie, is this the first time in aviation history that a software system instigated a fault that caused death and the pilots never even knew about the particular software system and it characteristics? (ie like they never knew about EMACs in the 737 MAX)
Nope, not even close to the first time. The Airbus A400M military aircraft took off with no calibration data loaded into the engine computers. This caused all 4 engines to shut off in the air, causing a crash. It seems the engine computers should have been programmed to do something rational, such as not even start, or not allow the engines to be advanced beyond ground idle.
I doubt these are the only cases.
Jon