Is this really a 4K projector?

No. The native resolution is 1280 x 720, and it accepts up to 1080p input. The 4K in the name is marketing, not a spec.

Does it have a battery?

No. It runs only from a 5V/2A USB power source, and the charger is not in the box. Plan on a phone charger or power bank.

The projector, a USB power cord, a remote with a button battery installed, and an instruction manual. No HDMI cable and no power adapter.

How big a picture can it throw?

The listing gives no throw distance or screen size figures. With a standard throw ratio and low LED brightness, expect a usable image only in a dark room at modest sizes.

Will it work with my phone or laptop?

It lists Android, Windows, iOS and Mac compatibility over HDMI, USB and VGA. iPhones and many Android phones need a separate adapter or cable, which is not included.

Only as a cheap, portable novelty for dark-room viewing. If you want a real home theater image, save for a branded 1080p projector with published brightness.