A first note: to prove the conjectures, one should first prove that there exists an infinite number of twin primes, and this is probably not for tomorrow.

 

For any p, of the three numbers p, p+1 and p+2, one is surely multiple of 3.

Every pair of twin primes p and q (without 3) is thus of the form 6x-1, 6x+1 (as p+1 = q-1 is even and multiple of 3, p,q,being primes and thus not multiple of 3; thus multiple of 6).

It follows that the sum of two twin primes (without 3) is multiple of 12, i.e., p+q = 12x.

 

The number of the known prime twin is larger than n/ln(n)^2 (this has been checked at least for n < 10^10), and summing two couples of twin  primes less than n (without 3)  we obtain more than n^2/ln(n)^4 multiple of 12.

As n^2/ln(n)^4 is much larger than n, as soon as n is large enough, most of the obtained multiple appears several times.

At the beginning some holes could appear (noted in point 2 of the conjecture), but after that, the probability for a hole is very small, and searching for a counter example is possibly almost hopeless as searching a counterexample to the Goldbach conjecture.

 

Regarding generalization 1, it can also be noted, that for all d non multiple of 3, the sum of two primes (different from 3) at distance d is also multiple of  12, and similar reasoning to the above could lead to similar conclusions.

 

Generalization 2 need more precision, as for example the fact that there exists infinite multiple of 12 imply that 4) directly follows from 2) (f=12, 24, 36,…).

On the other side, 1) is an immediate consequence of 2).