Does the LD2412 detect people who are not moving?

Yes. It is a presence radar, so it reports a still person as present, unlike a PIR motion sensor. Very still sleepers at long range can still drop out, so placement matters.

How do I connect it to an ESP32 or Home Assistant?

It talks over a 3.3 V UART. Wire TX and RX to the ESP board, power it, and use an existing LD2412 component or library in ESPHome or your firmware to read presence and distance.

What is the difference from the LD2410?

The LD2412 is a later module in the same family with updated firmware and configuration options. Both give presence plus distance; check the pinout and library support for your board before swapping one for the other.

Does it work through a wall or case?

It works through thin plastic and glass, which is why it fits inside a project enclosure. It will not reliably sense through brick, concrete or metal.

What comes in the box?

You get the sensor module itself. Power, a host microcontroller, wiring and any enclosure are not included and must be sourced separately.

Can it tell how far away a person is?

Yes. It reports the target distance over serial, so you can trigger lights or automations by zone. Accuracy is roughly in the tens of centimeters and depends on tuning.