I've got a lumbar radiculopathy and I'm wondering if this pain is for my lifetime or if it is treated the pain will go away permanently?

Varies. It varies based upon the cause, but most cases of radiculopathy are very treatable and so it typically does not have to be a permanent condition.
Will Improve. Without treatment it may be around for awhile. Naturally the radiculopathy pain improves by 3-6 weeks on its own. However if it persists after that you may choose to get care, otherwise discs typically heal over 6-9 months,
Probably not. That's just impossible to say. It depends heavily on what's causing the pinched nerve. Is it just a bulged/herniated disc? Is there some joint arthritis with spurs pinching the nerve? What about a synovial cyst? How big is the herniation if there is one? Epidural steroid injections can help tremendously for acute (new) radiculopathy, but will likely eventually need to be repeated.