Every soul is allotted 6 numbers, chosen at random from the positive integers.

If such numbers are relatively prime, then the soul is admitted to Heaven - otherwise, not.

What is the probability of an eternal reward?

What is the probability of an eternal reward, if such numbers must be pairwise prime?

Note: the integers 30, 42, 70, and 105 are relatively prime, while the numbers 121, 122, and 123 are pairwise prime, see the definitions at https://primes.utm.edu/glossary/xpage/RelativelyPrime.html

I agree with your idea for the second part.  Of note is that the 1/p term cancels if you expand and simplify