Have no idea what shape you are in so it may be that you need to work on your core strength. You can also employ tools like the fitball or just put a rolled up towel under the small of your back.
As far as exercises for fat loss around your abs - doesnt' exist -- you can't really "spot reduce" -- fat loss is about diet and on a tight diet you will drop bodyfat proportionally around your body, not in one particular place. Often for women the abs are a rough spot because estrogen causes us to store a greater number of fat cells in the waist / hips / thighs areas - therefore if you lose 2% bodyfat - you may look like you are cutting up on your upper body (as many women do) because your body naturally has a smaller number of fat cells in those areas than in your waist area -- there for 2% of a smaller amount of bodyfat will show cutting up more than 2% of a larger amount of bodyfat. Just keep workign on a tight, consistent & persistent diet & training program and the results will come.
go to a gym and get a trainer for a session... if they are any good, they can teach you a whole series of exercises for your core, that can be done while STANDING UP... this elminates the back issues BUT will also strengthen your back at the same time < ( that is, if they know their sheit)... good luck
your technique could be bad, that will also cause pain. best bet is to have someone that you feel confident that knows how to execute proper technique demonstrate to you exactly how to perform the exercises-go from there. if there is no pain, then it was bad technique.
My lower back always hurt when doing abs but having someone show me the proper technique it has been smooth sailing. As far as back extension, they will help you strengthen your lower back, but again, someone showing you would be best cause done improperly can hurt your back, as I have found out.