If the engine mounts are indeed in good condition (and it can be hard to tell just by looking) then you might be looking at a problem with the idle speed settings. Engines typically idle more slowly when they’re cold and in gear and there’s usually a compensation for this. In an electronically controlled engine like the Sirion’s, this idle-up function will be a job for the onboard computer. But if the computer has lost the plot or isn’t getting the right signal from, say, a temperature sensor, then it can’t bump the idle up a fraction when it needs to. And then, once warmed up, the engine seems normal.

A cold engine that spits a little water out the tailpipe is completely normal. Water is one of the by-products of the combustion process. It’s black because it mixes with soot in the exhaust system. But we only see it when the engine is cold, as once warmed up, the exhaust system turns the water to steam and we don’t see it. A black exhaust in a modern car that runs on ULP is completely normal also.

Unlike fuel and oil filters which are often located in easily accessible places and are able to be removed and replaced easily, most automatic transmission filters are located inside the actual transmission. Usually, this means removing the transmission's pan which is also usually the way to access the transmission's valve body and other parts that are part of a transmission service.

The best way to access this area is to have the car on a hoist and work from underneath.

Assuming you mean the Daihatsu's ignition timing, the on-board computer should control the timing. If this is out of whack, you'll find the car could be difficult to start, will idle poorly, use more fuel than it should, be down on power and may even overheat. Or all of those things. For reference, the correct timing for this engine is five degrees before top dead centre (+ or – two degrees) at idle. It can be checked by a workshop to verify whether it's correct or not.

If you're referring to the engine's valve timing, that's a different thing altogether. You need to reset the valve timing whenever you change the rubber toothed timing belt to avoid catastrophic damage to the engine. The valve timing of the car should not change between belt-change intervals, but if it has, then the damage has probably already been done. Again, for reference, the two camshafts need to be lined up with their respective timing marks, along with the crankshaft in its correct position before the new belt is fitted. This is not a simple job for a Saturday morning in the driveway.