Captain reported loss of situational awareness; allowing aircraft to descend below approach altitude in IFR; resulted in GPWS and ATC low altitude alerts.

Narrative

IFR flight to SLC. Weather conditions at SLC were reported as 1 mile visibility; Overcast 500; light snow. Cleared to proceed from QWENN Arrival at 11000 feet to HLMET intersection and commence the LDA 35 Approach. Mins of 4480 MSL were set into the FMS previous to commencing the approach.Tracked the LOC inbound and altitudes met at waypoints prior to waypoint WEEL. WEEL waypoint is 2 miles to Runway 35 threshold and altitude is listed at 4900 MSL. Just short of WEEL; aircraft inadvertently descended below 4900 MSL. Tower made low altitude call and asked if there was a problem and to climb; additionally the aircraft terrain avoidance system activated and called to climb. Correction was immediately made to climb back to 4900 MSL. Once aircraft was back at 4900 MSL; aircraft continued to VDP at which time Runway 35 was visible and uneventful landing was made.Once the aircraft was shutdown; PIC and SIC debriefed why descent below published altitude at waypoint WEEL happened. PIC had visual contact with the ground prior to WEEL and inadvertently allowed aircraft to descend below published altitude. Crew discussed importance of situation awareness and crew coordination throughout entire phase of flight to shutdown. Also discussed importance of flying a stabilized approach throughout to landing. Multiple lessons learned today.