Q. Power and storage consumption

There's no support for USB host (I think it requires extra hardware anyway), so you'd have to do something else. As others have said, SD card is probably easiest (in the next released version of the firmware), but then there are also several other types of EEPROM that will work and could potentially be a bit smaller/draw less power.

Using a Typed Array and saving would work, but save() itself takes maybe a second during which the Espruino is totally unresponsive (and then everything restarts), so you may well miss some data from the GPS during that time.

If you're confident in the battery then I guess you may not need to worry about saving at all - just let it run and pick up the data afterwards.

Using setDeepSleep(1) will set Espruino up so it enters low power sleep modes when it can. Right now that's still around 1mA because of some issues with shutting down the USB peripheral so it can be restarted again, but that will change in the next month or so when the new USB drivers get added, and we'll hit the microamp level.

For peak power, I can't remember the measurements but it should never go above 40mA. Potentially the SD card could add a lot of power draw though - honestly the best bet is probably just to measure it when you've got it all set up :)