I have the DX with the extended battery and I have purchased a couple of cheap cases and there was one that I got from Ebay that works with the extended battery and the battery cover. The reason why it works is because of the tabs to connect the top and bottom pieces.
I have found that if a case has a snap on tab then it will not work with the extended battery and the battery door. The one i have from Ebay has several clasp around the case. What I mean is both the top and bottom have peices that are vital to the sealing the case onto the phone. Others that i have tried only have the top piece or only the bottom piece with tabs that seal the case shut. That will not work with the battery door and the extended battery. Without you are fine.
As far as the life of the battery, I have never run out of battery life during the entire day. My mornings start at 6:40am and end between 11-12 midnight. So that is easily 16-18hrs of battery life with constant updates from Twitter, Facebook, Gmail, Text messages, Android Central, Endgadget 2 hours of Bluetooth music for the gym and miscellaneous listening, constant internet browsing and phone calls. When i charge the phone at night, I might have 20-30% left or 5-10%, but it has never shut off on my during the day