Smallest is 11 & Largest is 77.
The smallest prime number in which all of the digits are also prime is 2.The smallest multi-digit prime number in which all of the digits are also prime is 23.
You had me until "product." The product of 4 digits can't be prime.
The two smallest prime numbers are 2 & 3. Their product is thus, 2 x 3 = 6.
It is 27,000.
The smallest prime number, of course, is 2. The smallest prime number whose digits add up to 2 is, of course, 2. The next-smallest is 11. It is left as an exercise for the reader to determine whether the other numbers whose digits add up to 2 (e.g. 101, 1001, 10001) are prime.
The four numbers are: 113, 131, 151 and 191. For the product of the digits to be prime, the number must contain 2 ones - which greatly simplifies the exercise.
There are two numbers that satisfy the criteria. 38 and 83