"Payment Date" is when the buyer finished checkout with the payment processor; "Processed by E-j" is normally when we received confirmation of completed payment from the payment processor. However, if you're not selling digital goods such as downloads, you might just want to have us log orders as soon as possible; you can do this by going to Seller Admin > Payment Preferences, where you would uncheck the "Wait for pending payments..." setting.
If you download any log view to your computer, you can open that in a spreadsheet program like Excel and sort rows by any column you wish; however, I'm not sure how useful this might be for your order-fulfillment purposes. You might consider using our custom/3rd-party Integration feature to send the order data to a custom script at your end that would do whatever you wish with it -- e.g., save it to your own database where you could check off orders as you fulfill them:
http://www.e-junkie.com/ej/help.integration.htm