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@Gordon
Sorry I'm a bit late to this - not connecting GND is likely to be a big cause of the mismatch of data you're getting. Connecting GND together on all devices won't cause you any issues at all (in fact not connecting GND is far more likely to break things as you can end up really stressing the IO pins).
The Pico has a 3.3v output which should be capable of about 200mA sustained, 400mA peak - however I'd only use that for stuff that needs regulated 3.3v (eg. not a power supply).
The Pico also has a VBAT pin which provides somewhere in the region of 4.5v when from USB or battery voltage when powered from battery, so I would use that: https://www.espruino.com/Pico#pinout
There's even a self-resetting fuse in there so if you draw too much power (>500mA sustained) from USB it'll just temporarily cut out without breaking anything.
It's entirely likely that the sensor will function perfectly fine from 4.5v without the boost converter - but if you're planning on switching to LiPo power and some point then you'd probably want the converter. When on LiPo you can just connect the LiPo power to BAT_IN and the Pico will automatically power from USB when USB is connected, or LiPo when USB isn't.
Just a note: The baud rate specified in the Web IDE will have no effect for you on the Pico, since the Pico is connected by USB.