It seems pretty clear from these symptoms that the engine is massively overheating. That explains the boiling water in the overflow tank which will be dirty mainly form rusty deposits within the engine’s cooling system. The question then becomes what’s causing the overheating.
There are lots of reasons for an engine to overheat, including low coolant level, a blocked radiator, collapsed radiator hose, failed head gasket, incorrect ignition timing, poor fuel mixture, a damaged or missing fan shroud, slipping fan belt, worn water pump and many, many more. But a good mechanic should be able to assess the situation, do a few quick tests and determine the root cause of the overheating, then you can move forward on a fix, knowing that you’re tackling the correct issue.
The other piece of advice is to not drive the car until it’s fixed. Continuing to drive with an overheating engine is a great way to destroy the engine completely. At that point you need a new engine or a new car.
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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.
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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.
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