What kind of doctor deals with spine problems and the subsequent nerve problems that arise from them?
Orthopedist.
Possibly with a Nurologist consult?
Do you have a General Practitioner (GP) or Primary Care Physician (PCP) that you normally go to first? He/she should be able to refer you to the appropriate specialist in your area.
For surgical options, eiither a Neurosurgeon or an Orthopedist who does spines (most don’t and even fewer do cervical (neck) if that’s your problem). For non-surgical options, your primary doctor can probably manage with a physical therapist referral.
I’m having occasional extreme pain on the sides of my thighs (due to sleeping in a certain position on a too-firm mattress) and probably-not-coincidental low back pain (in the area of L4/L5, which I know has a nerve path that follows that thigh area).
I have the ability to seek out a specialist without a referral, and I’d like to avoid paying for the doctor visit just to get a referral. I realize that paying to get a proper referral is exactly why I would be paying (ie I’m not complaining), but I’m on a budget and hope I can get myself pointed in the right direction on my own.
(I’m not asking for medical advice :))
An orthoaedic physician or neurologist handles spinal injuries (or a physiatrist in some cases). A neurologist would be the best choice for an injury with radiculopathy.
Good thing you are not, because he would tell you to change mattresses, or get a soft topper and the classic joke - Doc, it hurts when I do that, well stop doing that.
Also you can sleep with a shaped pillow between your knees to hold your legs in position.
If you know that how you are sleeping is causing the problem, you really don’t need the advice of a bunch of strangers on a message board, you know you need to change something. The easiest things to change are the mattress firmness and your position.
My mattress is fine and I never have a problem sleeping on it. It’s other people’s mattresses: my dad’s, my sister’s, the woman for whom I house sit. I can’t really control sleeping arrangements outside my own house, much as I would love to! 
I start out with pillow between knees, hugging pillow, neck well supported on pillow. How I end up during unconsciousness is also something I also can’t easily control.
