Is it warm enough for freezing winter days?

Yes for typical cold city weather, especially layered over a sweater. It is a down-filled coat, so it holds warmth well, but it is not rated for extreme cold or long exposure.

Can I wear it in rain?

Light snow and drizzle are fine, but the shell is not fully waterproof. In steady rain the down can get damp and lose loft, so carry an umbrella or a shell.

How does it compare with a short puffer?

It covers the hips and upper thighs, which adds warmth on windy commutes. The trade-off is more bulk and less freedom of movement than a hip-length jacket.

Yes, it comes with an attached hood. It is not a helmet-compatible mountaineering hood.

How should I wash it?

Use a down-specific detergent on a gentle cycle, then tumble dry low with dryer balls to restore loft. Avoid regular detergent and high heat.

Is it good for hiking?

It works for light walks, but it is a casual coat. For active hiking in wet conditions, a waterproof shell with a synthetic or down midlayer is a better choice.