What is this component? (DMP2035)

The MOS FET is very low resistance when turned on, i.e. 5v is missing on the left side of the drawing.

Battery voltage will be seen at the left side of the switch (pin 3).

When 5v is present, probably form USB, the FET is off, battery not connected to the switch.

D3 prevents battery voltage from connecting to 5 volts.

The 3.3 volt regulator will get input voltage from battery (through Q1) or 5v (through D3) or VCC_IN