Which are correct:
1-The animal died with a lot of pain.
2-The animal was killed with a lot of pain.
3-They killed the animal with a lot of pain.
I think "1" is correct but "2" and "3" might seem to be saying that "pain" was used to kill the animal. I am not sure about them.
Maybe this is better:
4-They killed the animal painfully.
I'll assume you mean the animal was in pain.
You can say:
The animal died in a lot of pain.
The animal suffered a lot [of pain] in dying.
They caused the animal a lot of pain in killing it.
They made the animal suffer as they killed it.
Killing the animal caused it a lot of pain.
Klling the animal caused it to suffer as it died.