There are several schools of thought about the rapture.
In all of history, we only know of two people who were raptured: Enoch and Elijah. Everyone else lived and died including Jesus.
The 3 present day theories about the rapture center around the 7 year tribulation period. Your cousin is a pre-tribulation believer, but there are others who believe in mid tribulation, and post tribulation raptures. Read 1 Thessalonians 4-7, for a Biblical reference, but there is no indication of when this occurs.
There is still a serious question about all of this. Who are the believer that are to be raptured? Those who simply believe that Jesus is Lord? Most like not, because Jesus says that even the demons believe that. So we must be talking about the true believers who follow Jesus and His teachings about living in the heavenly will of God, where there is no suffering and death, and these people do everything in their power to bring that heavenly will to earth and free creation from its present corruption. And, if this is true, then God in all His grace will rapture the animals along with the people who care about them, and leave the corrupt people behind without the animals they love to kill and eat.
We hope this helps.
In the Love of the Lord,
Frank and Mary