Can I charge and scan simultaneously?

Yes, with the included USB power adapter (or a phone charger) -- but not when connected to your computer via USB.

Doxie is a standalone scanner -- that is, it's not designed for use while connected to your computer via USB. Doxie can't write to its memory if your computer is also accessing it. Like a digital camera, Doxie works while disconnected -- you scan, then download at some later point in time. So, don't scan while connected to your computer. If you do so, your computer thinks you're continuously unplugging and reconnecting the scanner.

To charge and scan simultaneously, use Doxie's included USB power adapter.

I lost my power adapter

You can use any USB/AC power adapter to charge Doxie Q.