I talked to a Sony CSR when I was making a 2nd account and asked if a character would keep vet rewards if I moved it to new account. answer NO.
I asked if another character could claim the vet reward if the character that has them is moved to a new account. answer NO.
Once a vet reward is claimed that is it, it has been claimed and cannot be 'un-claimed'.
Yes and no. The key here is the usage of NEW. If you claim a vet reward and move the char to a brand spanking new account, then no, it won't have any vet rewards. If you wait a year, it will have lessons, and you will have one claim on that account.
An example:
I have 3 accounts, 1 9 y/o, 1 6 y/o, and 1 3 y/o. I have 9 claims on the first, 6 on the second, and 3 on the third.
I mistakenly make a cleric and a warrior on the 9 year account and claim vet rewards on both. I choose to move the Warrior to the 6 year account. He now has access to the first 6 vet rewards, losing the last three. I can still claim 6 vet rewards on that account.
A week later, I decide to be naughty and visit Zipp's site and make a monk, a rogue, and a berzerker on the 3 year old account, claiming vets on all three. After my fun, and knowing the ban hammer is incoming, I /charmove the monk to the 6 year account, and the other two to the 9 year account.
The monk ends up with 6 vet rewards total, and the rogue and zerker get 9 each. I still have 6 claims on the 6 year account, and 7 claims on the 9 year account.
Veteran reward claims stay with the character. If you move the character to an account with less, or no veteran rewards, you lose them until the first year has passed. If you delete the character the claim is gone forever.
Oh and the example given above? Is not hypothetical, although the site wasn't Zipp's at the time.