On my Zoom H2, I use a SDHC card that isn't on the list of "approved" cards. It works fine. And I agree with Ray - if you can plug an SD card into your computer, the file transfer will be faster and safer than using the USB cable. Looks to me like the H1 uses a power adapter that connects via USB cable. I'm guessing that means it will work on phone chargers that use a USB connection.
The H2 has a setting for using rechargeable batteries, and I suppose the same will be true for the H1 - so I suppose you will be able to charge your batteries while they're in the recorder.

-Joe-