Ofcourse it 'starts fast' when you leave the receiver fully powered. gpsapp doesn't turn the receiver on or off, so the receiver just keeps tracking the satellites. The interesting part is what happens when receiver comes up from nothing. i.e. even the internal battery is drained. Does it remember the coordinates in that case?

Measuring current is can be done by measuring the voltage drop over a known resistance. Assuming a current draw between 100 and 250 ma, the voltage over a 5 ohm resistor in the power feed should drop between 0.5 and 1.25 volts. Most resistors should be able to handle the 0.4W of dissipated heat if I'm at all correct with my estimates.