Not quite true: the first twin primes, 3 and 5, surround 4, not divisible by 6. However, for any other twin primes, both will not have factors of 2 or 3, so the number between them must have a factor of 2 and a factor of 3 (the latter by the same logic as in the parent post: otherwise three numbers in a row would not have a factor of 3).

Arghh. True.

Any pair of consecutive odds both of which are not divisible by 3 surround a multiple of 3

5 and 7?

those surround 6.