Probably methane or propane released into the bottle slowly so it falls to the bottom and pushes the air out the top until the bottle is basically nothing but the fuel gas.
It's called a "whoosh bottle", and the fuel used is usually a volatile alcohol like ethanol or isopropyl. My guess is because alcohols are easier to pour in than LPG and release a little less energy when they burn.
The second flame at the top of the bottle is because space in the bottle is oxygen-deficient, so there's lots of unburned fuel that gets pushed out by the heat of what is burning, and it can't ignite until it's outside of the bottle, where there's plenty of oxygen.
The increased mass of the large stone will carry it farther against the drag of the water's surface tension. At least, that's my guess, followed by crying "shenanigans!"