Is DBL_MIN the smallest positive double?

I'm restricting this answer, perhaps unnecessarily, to IEEE754 floating point.

DBL_MIN is not allowed to be a subnormal number.

But std::nextafter is allowed to return a subnormal number.

Hence the return value of the latter could be less than DBL_MIN.

For more details see https://en.wikipedia.org/wiki/Denormal_number

Is DBL_MIN the smallest positive double?

Not certainly.
DBL_MIN is the smallest positive normal double.

DBL_TRUE_MIN is the smallest positive double (since C++17). It will be smaller than DBL_MIN when double supports subnormals.