I have a cut on the bottom of my foot. I also have dry skin on the soles of my feet and they can get very cold. Could this be because of my diabetes?

YES. You're young. How long with diabetes? What's your A1C-should be below 7. Cold feet, dry skin could be what's called autonomic neuropathy, associated w/ diabetes. Make sure you see your PCP as indicated, and a podiatrist for that cut.
If u have a cut. On the bottom of your foot and are diabetic I suggest you see a podiatrist first and foremost. The dry skin may or may not be a result of autonomic neuropathy. The cold feet may be neuropathy as well, you may feel the feet are cold and they may not be. If they are it could be Raynaud's. I would doubt a 17 year old would have peripheral vascular disease ..,yet......