Does moderate sun exposure (10-15 minutes) without sunscreen have carcinogenic effects?

Not Likely: Probably not, unless you're an albino. And that's the amount of time each day suggested to get your Vitamin D going. But wear a hat or sunblock on your face. You just need part of your body exposed, preferably after 2pm After that, use sunblock everywhere (except inside your nose and between your toes...)

Any sun exposure: Has some carcinogenic potential related to UV exposure. But remember its exceptionally small with limited exposure like you mention. There are many benefits to sun exposure too, you level of exposure just needs to be balanced based on your risk for developing skin cancer.