Voter roles are registration are effective. I vote in almost every election, and they can tell my ballot status when I go in. My vote is tied to my address and other factors, so it seems like it would be clear if I wasn’t a legal citizen.
I also have to show a government ID when I vote, but it doesn’t seem like that’s made a big difference in the security of the elections I participate in.
Depending on the company, you often have the option of either a group or a private cremation. Group cremation is what you described, but private cremation ensures you only get your pet’s ashes returned. The company my vet uses even offers the option for you to be present and view the cremation.