Does it actually cool the air?

It cools by moving air over your skin and by adding a fine mist, which feels cooler as it evaporates. It does not refrigerate the air, and its effect is limited to roughly one square meter around you.

How long does the battery last?

The listing gives no runtime figure. Expect the longest use on the low speed and the shortest on the top speed, since higher fan and mist settings draw more from the small lithium battery.

Can I use it without water?

Yes. The fan runs on its own; the mist is a separate function. Run it dry when you only want airflow and to avoid dampness near papers or electronics.

What comes in the box?

The fan, a USB charging cable and a user manual. No wall adapter is included, so you need your own USB power source to charge it.

Is it good for a whole room?

No. It is a personal device rated for about one square meter. For room cooling, a full-size fan or a proper humidifier does the job better.

Wipe the exterior and keep the mist outlet clear, and empty the water reservoir when it will sit unused. Follow the manual's care section for the reservoir to avoid mineral buildup.