I realised that I was lacking in knowledge on how the Jazz fuel injection system actually works, so decided to plug in my Foxwell NT530 scanner and see what it would tell me.
Warmed the car up fully with a few miles drive and then plugged to see what was happening.
Firstly, I'm sure you're right that there is no Mass Air Flow sensor on the Jazz, only an Manifold Absolute Pressure sensor, which I believe works in conjunction with the throttle posistion sensor and the "pre-cat" oxygen sensor to determine the amount of fuel to inject.
The MAP sensor reads "absolute pressure", and as standard atmospheric pressure is around 100 kpa, anything less than that value indicates a vacuum. With a closed throttle, the engine running and the pistons sucking, this gives the MAXIMUM vacuum, which equates to the MINIMUM kpa value.
With a coolant temperature ranging from 84 to 90 during the test (the fan was starting to come on at times).
At tickover (680 rpm ish), the kpa value was ranging between 24 to 27 kpa.
At 2,200 rpm, the kapa value was ranging between 21 to 22 kpa.
At 3,000 rpm, the value was fairly constant at 23 kpa.
The highest kpa value I saw was a brief 34 kpa as the throttle was being opened.
My absolute throttle position ran at around 12-13% without me touching it, and would rise to around 20% when I put my foot gently on the throttle to raise the revs.
The "pre-cat" O2 sensor was showing the classic wave form and was varying from 0.11 to 0.7 volts over a period of around several seconds. It looked very much like your O2 sensor graph that you attached, so I would doubt if that was your problem.
The "post cat" O2 sensor was very constant at 0.67 volts - which is what you would expect as it shows the cat is doing its job correctly (this O2 sensor is purely there to determine if the cat is working correctly and isn't used to control the injection as far as I know. You say that your post cat sensor is reading "0.45-5 V) .... do you mean that it's between 0.45-0.5 volts or is it actually going to 5 volts. If it is going to 5 volts then that's very suspicious??
Short and long term oxyen trim levels were small (though stupidly, I forgot to take a note of the readings)
Good luck