Twin, cousin, and sexy primes are of the forms $(p,p+2)$, $(p,p+4)$, $(p,p+6)$ respectively, for $p$ a prime. The Wikipedia article on cousin primes says that, "It follows from the first Hardy–Littlewood conjecture that cousin primes have the same asymptotic density as twin primes," but the analogous article on sexy primes does not make a similar claim.
Q1. Are the sexy primes expected to have the same density as twin primes?
Q2. Is it conjectured that there are an infinite number of cousin and sexy prime pairs?
Q3. Have prime pairs of the form $(p,p+2k)$ been studied for $k>3$? If so, what are the conjectures?
Thanks for information or pointers!