Sit-ups are profoundly inadequate "ab" exercises for someone with back problems. Please don't do them.
Are you going to physical therapy? They will teach you ab exercises that are functional (planks, trunk stabilizations, posture work). They will also teach you how to stretch. Tight hipflexors can pull your vertebrae around, so keeping them the proper length is very important.
As far as biking is concerned: we encourage our back patients to bike. Your PT will check your positioning on your bike and check your posture.
When I had an MRI I asked for a valium. Claustrophobic. They let me bring in a cd, so I listened to my own music while I was in the tube. It was fine.
Remember that high-velocity chiropractic treatments stretch out the ligaments that hold your back aligned. As your postural muscles get stronger and you hold yourself in proper alignment more often, those loose ligaments will gradually tighten up again to a functional length and you will have less "going out" in your back. It's all about using your postural muscles to maintain the alignment you and your practitioner achieve! (I hope your chiropracter wasn't the one who told you to do sit-ups)
If you have a disc issue, unless it's a huge rupture your doc will likely send you to PT first. Be diligent with the PT exercises, as they can prevent your needing surgery (which is why docs send back patients to PT if there is a chance PT will work).
Generally back surgery is a simple procedure these days. I've seen patients with tiny little scars!
Keep us posted!
"If Americans want to live the American Dream, they should go to Denmark." - Richard Wilkinson