If it is ever proved that there are infinite twin primes (like 3,5 and 41,43) will this imply that there are infinite primes separated by every even number?
It would depend on how that proof is organized. A proof could (theoretically) be devised which proves purely twin primes. A more general proof could (theoretically) also be devised, but that would be a separate thing and, it’s reasonable to assume, much more difficult.