Can this amp power a subwoofer?

Yes, if you bridge two channels. Bridging pairs channels into one stronger output, so a four-channel amp can run two speakers plus one bridged sub, or two bridged subs.

Does it need a separate power cable?

Yes. It needs a fused power wire run from the battery, a ground to bare metal, and a remote turn-on wire from the head unit.

Is Class D good for full-range speakers?

Class D is efficient and fine for speakers and subs. Early Class D designs had a reputation for harsh highs, but modern ones are clean enough for most listeners.

How is it different from a name-brand amp?

Name brands usually deliver closer to their rated power, include better crossovers and protection circuits, and have clearer manuals and warranty support.

What speakers can it drive?

It is meant for standard 4-ohm car speakers. Check the minimum impedance for bridged mode before wiring subs, since low-impedance loads can trigger protection.