Question 1 comes up a lot. For example, it was discussed on sci.math back in 1990. It was also discussed in print in the following article:
Quotients of primes, by David Hobby and D. M. Silberger, Amer. Math Monthly 100 (1993), 50–52.
More recently it has shown up on Yahoo Answers and math.stackexchange.commath.stackexchange.com.