How can I install this programmable switch?
Turn off the breaker, undo both wires from the switch, then turn on the breaker. Get a NCVT (non contact voltage tester) and see which of the wires is hot. That will be the one back to the panel, the other one will be the load.
Yes, just connect the white to the other neutrals in the box, so they are all wire nutted together.
The programmable switch requires a neutral because it needs current itself all the time.  If you just had the 2 black wires, there would be no current flow through the switch when it was off, so no way to power the electronics.