Yes, this is normal to have an increase in back pain after some weight gain. Adding a little weight causes additional strain on the muscles and joints in your back. The stress can put the low back out of alignment, which any chiropractor should be able to help you with.
Due to wear and tear or injury, your joints and the bones in your spine get pulled and twisted out of alignment, putting pressure on nerves. A chiropractor's job is to put those joints back into place through an adjustment. A physical therapist works to rehab and provide guided exercise to stabilize and improve muscle function. These two professions work well hand-in-hand, but if your doctor suggested a chiropractor, it wouldn't hurt to try that first. The rest of your body functions off of the nerves coming out of your spine, so removing that interference would be the key place to start.
Yes, there are certain exercises that would help with gout. Simply moving your joints in circles would help. You can do yoga, swimming, and elliptical to exercise but keep pressure off your joints. Stretching is good, too.