If we fly out of PSP and have a connecting flight in San Francisco to go to Honolulu does the United check in counter send our check in baggage directly to Honolulu?

Good One...,I Think it depends on who sends the luggage.., You may have recheck them in if There's a long lay-over at the airport.

Yes if your flight is a non stop but if you have to change planes then your luggage is automatically transferred to your connecting flight or flights

Yes, your bags go through automatically..I recently flew psp to Amsterdam, with 2 stops.. I didn't see my bags again until I got to Amsterdam