The battery will most likely not explode, but just ignite. The melting of the chemicals and metals just gets really really hot, so anything else around it will start to burn eventually. So don’t treat it like a bomb, more like a very hot iron. If you can, find a temporary spot for the battery. Maybe in a garage or basement. If also possible, use a metal container. Dirt/sand is also a good option.

If the battery is quickly removable I would put it into a sand bucket and cover it with sand.

I actually had to check some resources myself, as I was unsure if it was really useful in that case. Those blankets usually help stopping a fire by limiting the amount of oxygen that gets to it - without oxygen, no fire. Unfortunately, many batteries have oxygen in them, not much, but enough to keep it going. So the fire won’t stop in that case. But what the fire blanket does, is give a layer of insulation, thus reducing sparks flying around and reducing the temperature directly above it.

Fire blankets are always a very useful tool, as they are easy to use and at least protect the person holding it (in small fires, obviously). If it doesn’t help, it does not make it any worse.

Just send them the pic and say you discarded it for safety reasons. Maybe take another pic with something identifying like the original receipt or something unique to the purchase and a date for proof. Make sure to get a clearer pic of the bar codes on the top as well.

Knowingly mailing a swelling battery is a safety hazard and there should not be any issue with disposing of it as long as you have something to show it was the one in that particular laptop.

I would suggest opening a case with the manufacturer first if its still under warranty, that way they should be able to get you a replacement even after you discard the faulty battery