If you search the threads for "replacement battery" you can find several discussions and opinions from members about their favorites. You'll find recommendations for batteries from PPC techs, Mugen, Boxwave, and others.
You're right in your assumption about the external battery power. Have you tried Settings>System>Brightness>External Power settings?
For screen captures, I've used add-on software. Have you read Gigi's superb "What Can I do with my Axim?" tutorial. That is a great resource for ideas.
http://www.mobilitysite.com/boards/v...o-with-my-axim