No you can't boot both at the same time.
To move files from one to the other (say lenny to squeeze), edit the profile for squeeze and add the root lenny drive as another parition, boot into the squeeze profile and mount the lenny drive, then you can simply copy the files using cp.
Switching websites shouldn't be a problem if they're both the same configuration they will also share the same ip, so as long as your webserver configurations are the same it should work fine.
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