Right, I understand now.

Firstly, there is something quite amiss if your idle adjustment screw is set to its maximum to get the bike to idle and start. I think this is indicating other problems and you’re just trying to compensate for them with the idle speed?

You can set the TPS, but ideally you set it after you have got the engine idling correctly. The procedure is to set the idle first then loosen the screws on the TPS body so as it can be turned in its slots and rotate the TPS until 0.63 – 0.73V is detected yellow and blue/black wires in the back of the connector. Tighten the screws when the voltage is within spec. All this assumes a good TPS is fitted (which you say has been replaced).

However, as I mentioned, you really need to have a stable and correct idle speed first (1400-1500 rpm) – which you haven’t got. This, along with the backfiring through the intake, leads me to think that your valve adjustment is not within specification – probably the intakes valves are set too tight and are not closing fully after the intake stroke has completed? This may cause a small amount of fuel/air mix to be forced the wrong way through the intake and would explain the troublesome idle, hard-starting and backfiring.
You also need to have the CO set to somewhere around 10-15. (It’s not exact, but roughly what it should be for your year of bike). Also check the throttle angle is between 15 and 20 in the diagnostic mode. If that is way out then you’ll have problems.

Before going any further though I would take the tank off and check the cold valve clearances. Then go back to setting the idle, TPS etc. You can also check the valve/cam timing while you’re at it which, if not right, will cause similar symptoms.

It is also worth carefully checking for intake vacuum leaks as well as these can have serious effects on idle and slow running. The rubber intake hose on older bikes is notorious for perishing and leaking vacuum.

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