Prime Problem

Take some time to read the statements below.

$[1]$. It is impossible for $p$, $p+2$ and $p+4$ to be all prime numbers where $p$ is a prime number greater than $3$.

$[2]$. It is possible for both $8p-1$ and $8p+1$ to be prime when $p$ is a prime number.

$[3]$. If $p$ is a prime greater than $3$, then $p^2-1$ is always divisible by $12$

Which of these statements are true?

Note: This problem is a part of the set "I Don't Have a Good Name For This Yet". See the rest of the problems here. And when I say I don't have a good name for this yet, I mean it. If you like problems like these and have a cool name for this set, feel free to comment here.