I am creating a google banner for Double Click & assigning a backup image to it but I am not sure if the banner border is going to be applied by double click or should my backup image have a border?
To be honest I've never had anyone check the backup image for borders or not so I think you could get away with it.
To be safe I think adding a border is a good idea and it doesn't really hurt anything.
My understanding is that either the backup image is shown OR the HTML5 banner is loaded. Your backup image doesn't get loaded inside of the HTML5 banner so if you're counting on the border of your banner to sit on top of the image, that will not work. I'm not 100% sure of that however.