Yes, I've added a charge pump to boost the voltage above the undervoltage lockout threshold for the startup period. The simulations seem to run fine now. I wish they would lower the threshold to 3V or below (a discharged 3.7V lithium-ion battery).Since I have no clue as to what a DGD05473 actually is, what I can tell you is that the short answer is NO! Various circuits have minimum voltages and below that minimum voltage nothing happens, at least nothing useful.
Yes, it's that part. It seems to be quite popular. Used in Teslas, for instance: https://www.chargerlab.com/teardown-of-tesla-model-y-15w-wireless-charging-module-2/