No. The weight gain of pregnancy, the curve that develops in the lower back with pregnancy, prolonged pushing, etc these are all likely to cause back problems after delivery. The epidural is probably not a major reason for it. As soon as you are cleared by your doctor start exercising. Yoga, in particular, is very good for strengthening the back. Also eat a diet of largely fruits and vegetables.
No. Generally there can be some transient back pain after an epidural injection. I would suspect it would be something else causing your back pain if it does not let up. However if you are concerned or have other issues like increasing numbness or weakness that you see a doctor soon and maybe get an MRI if warranted. Rarely hematomas are known to happen and should be evaluated for if concerned.
Unusual. It is possible that am epidural can cause prolonged back ache, particularly if it was a difficult placement. It is a little more common to have discomfort at the insertion site. Consider other causes, like labor, pushing, and hyper extension of the lower extremities during the second stage of labor.