A twin prime theorem, and a reformulation of the twin prime conjecture

Let $n$ be the sequence of integers such that $(6n-1,6n+1)$ are twin primes. For your first question, at the very least such an observation has been made here by "Jon Perry" here: https://oeis.org/A002822

You'll find numerous other observations within that link as well. In particular, there are plenty of red-herring reformulations of twin primes, another one being $(4(p-1)!+1)\equiv -p\pmod{p(p+2)}$ iff $(p,p+2)$ are twin primes. I say red-herring because the actual usefulness of these is dubious. In your case, showing a number is not of the form $6ij\pm i\pm j$ is considerably more difficult than it looks.

Concerning your second question, you're effectively asking us to tell you how to prove the twin primes conjecture. The closest thing I can think of is general sieve theory, for which a good introduction can be found here.