Mazda 3 Engine Problems

You need to measure the idle speed and compare it to the idle speed spec. You can do that using a tachometer. You can check the fuel economy over a tank full of fuel and compare it to what you should expect. Mazda's official claim for the Neo was 8.9 L/100 km and I would expect your car to be around that mark.
 

The feature is only really effective in heavy stop-start traffic where you can save fuel by not idling the engine while you're stopped. If the traffic isn't stopping and starting regularly the system doesn't have much impact on the economy. I would expect carmakers to beef up the starting system to ensure the more frequent use of components like starter motors doesn't lead to their premature demise. I would also expect them to have thoroughly durability tested cars with the feature to ensure there are no hidden issues of the sort you raise. On the cars and trucks I have driven with the feature I have always been concerned about the restarting when you need to move off, and you would want any delay to be minimal. The "gung-ho racing driver" behind you has the responsibility to avoid running into you, but I understand your concern.

Our LPG industry specialist says there is no reason that the normal Mazda scheduled maintenance of your car can't be carried out by a regular dealership mechanic. The only time an accredited LPG technician is required is when the high-pressure side of the gas system is to be opened for some reason. The dealer's behaviour is strange given that they are normally quite happy to sublet the LPG work to an accredited LPG business. Your vehicle would have been fitted with an AAFRB compliance plate when the system was installed, and the very presence of that plate is confirmation that the system on your car has been approved. If you are concerned about its approval check for the plate or you could contact the AAFRB at http://  www.aafrb.com.au to seek advice.

Normally I would suspect the valve lifters based on your description of the noise, but I would have another mechanic listen to it and diagnose it for you.

The alternative fuels people have told us that it is not correct. All petrol companies must display notices on pumps when ethanol is present in the fuel.

A failure of a water pump shouldn’t cause major internal engine damage. I would expect the failure to cause the engine temperature to climb and the warning light to come on. Providing you address the problem and don’t drive the car to destruction there won’t be any serious damage done. But I would be concerned that you have failed three water pumps in such a short time; that is not normal, as you appear to have been told. Have the dealer explain to you why he believes it is normal, and justify to you that it won’t continue to happen.

The diesel and petrol engines in the current Mazda3 have timing chains.

I'M NOT aware of a widespread problem, but perhaps other readers will let us know. As for the methylated spirits, yes, it will break up water in fuel. It's an old remedy.