12v Power adapter IC circuit design High voltage AC
That blue 222 2kV capacitor should be "Y-rated" for safety reasons and it is often called the "Y-capacitor".

It is there for EMI purposes (to lower EMI emissions which can disturb other equipment) and closes the path for the high-frequency signals generated by the switching chip or transistor at the primary side. Without this "Y-capacitor" many supplies would likely fail FCC and other regulations.

1) What the unmarked IC is, is any one's guess. There are many of these chips available and they're all slightly different but do operate in roughly the same way. Feel free to compare all application diagrams off all these ICs with your schematic but personally I do not see the point of that exercise.

2) Very likely this is needed for drawing a small amount of power from the transformer which is then used to provide power to the main IC. Your drawing of the transformer is not accurate enough to properly display how this works.

3) the IC is not connected to V+ as there could be 400 V on it and that could fry the chip. What makes you think that the chip needs the 400 V anyway ?

4) Reliability is a tricky subject. How would you distinguish between a reliable circuit and one that is not ? No circuit lasts for ever. If used properly most ICs are guaranteed a 10 years lifetime. If you mistreat the circuit, keep it too hot for example, it will break sooner.

AC power adapter design is a topic for experienced EEs. I fail to see why you would need to fully understand this circuit. If it is a very cheap one then you cannot expect it to last very long. As always, you get what you pay for.

I suggest watching some of the videos from Bigclive on Youtube, in many of these he takes small power supplies like this one to bits while he comments on their design.