The notion of the rapture is not mainstream Christianity. It is in the fringe. Gees, with over 1500 versions of Christianity in North America, you can't tell your sects without a programme!
Sure you can, and it's even quite easy, ask them which passage best supports their position. If they reference a person (he said) compared to an (in context) passage their doctrine is most likely false.
Anyway, back to the topic:
so what happens to our pets when rupture comes? Will they be as arbitrarily picked as humans?
Matthew 24:40-41, "Then shall two be in the field; the one shall be taken, and the other left. Two women shall be grinding at the mill; the one shall be taken, and the other left." I can see people wearing bright neon signs, saying, "Pick me! Pick me!"
Your quote is not a rapture reference, it is a gathering for destruction in the feast God prepares for the Eagles and the Beasts of the Field. Little errors like that can have you wishing for something that you really don't want. At the time the above verse is unfolding the safest way to get to Christ is to
walk and take in all the destruction along the way. Look carefully at the clothes worn by those being (air-lifted), you might see the same near the place that this will also be seen.
Jer:18:16:
To make their land desolate,
and a perpetual hissing;
every one that passeth thereby shall be astonished,
and wag his head.
La:2:15:
All that pass by clap their hands at thee;
they hiss and wag their head at the daughter of Jerusalem,
saying,
Is this the city that men call The perfection of beauty,
The joy of the whole earth?
Zep:2:15:
This is the rejoicing city that dwelt carelessly,
that said in her heart,
I am,
and there is none beside me:
how is she become a desolation,
a place for beasts to lie down in!
every one that passeth by her shall hiss,
and wag his hand.
As for pets and such they will not live much past the feast mentioned above. The battle with Satan is not over with the end of that feast, and even greater threat comes when he is released 1,000 years later. At that time all the people who have been alive are taken to Heaven and anything left on the Earth is sent to the lake. That is why 'other flesh' does not enjoy a splurge in numbers, etc. Other flesh (than man) cannot enter Heaven because they do not know about good and evil so they are not going to be judged. After that final judgment New Jerusalem descends to the Earth and the living waters flow out onto the new earth, that is when all other flesh (including pets) are given a new life, which also happens to be eternal. Hope that short explanation helps you understand how that frictional book actually works. The safety described below is death, the promise is that they will be remembered, and restored, by God.
Ho:2:18:
And in that day will I make a covenant for them with the beasts of the field,
and with the fowls of heaven,
and with the creeping things of the ground:
and I will break the bow and the sword and the battle out of the earth,
and will make them to lie down safely.
