Is this coat warm enough for real winter?

Yes for typical winter city weather. Down padding traps heat well, and the water-repellent shell blocks wind and light precipitation. In deep cold or long exposure you would want a base layer and a midlayer.

How does it compare with a standard puffer jacket?

It is trimmer and more technical. The stretch inserts and adjustable hood make it easier to move in than a boxy puffer, while the down keeps comparable warmth. A basic puffer is bulkier but often simpler to care for.

Can I wear it in rain?

Light rain and snow are fine; the shell is water-repellent. It is not a waterproof membrane jacket, so sustained heavy rain will eventually soak through. Carry an umbrella or a shell for wet days.

What does the double zipper do?

It lets you open the coat from the bottom as well as the top. That helps when sitting, driving or walking uphill, and it vents heat without unzipping the whole front.

How should I care for the down filling?

Wash it sparingly with a down-specific detergent and tumble dry low with dryer balls to restore loft. Avoid fabric softener and long soaking. Store it loosely rather than compressed.