I’m not Jewish, I have a question about touching the carcasses of dead non-kosher animals

As the title states, I am not Jewish, so sorry if this is a question about an obvious thing.

So, Jewish people can’t touch carcasses of animals that aren’t considered kosher or else they’ll become unclean, right? Then what would happen if, for example, a Jewish person had a pet rabbit and it died? Would they have to take the body away with pasta tongs? Are gloves ok since you’re technically not making contact with the body? Or would you call a non-Jewish person to take the carcass away?

And what if a kosher pet died on Sabbath? Like you have a pet chicken and it dies right on the living room. Would you need to wait until Sunday so you could take the body away or is it not considered work?

Sorry if it seems dumb, but I genuinely want to know. I know most Jewish people don’t have pets, but let’s say this is hypothetical.