These look identical from the outside, and most cans thrown away early are cold ones. A can that has been worked hard for ten minutes has a soft stream and a chilled tin, and both come back.
The test is patience. Stand the can upright at room temperature for a quarter of an hour and try again. If the stream returns, it was cold. If a rested, room temperature can gives you almost nothing, it is done.
A can that still has pressure in it is a pressurized container whatever the label says about being spent, and that is what decides how it can be handled. Standing it upright and pressing the trigger in the open until nothing more comes out is a minute well spent.
Do that outdoors or by an open window, with the nozzle pointed at nothing in particular. What is left is the tin, the valve and the trigger.
Do not puncture it, do not put it in a fire or an incinerator, and do not crush it. Those are the instructions printed on the can, they apply to the empty one as much as the full one, and they are the reason a duster does not belong in a bag headed for a compactor.
It also does not belong in a hot car, on a radiator or on a sunny windowsill, full or empty. Heat is the one thing a sealed steel container has an opinion about.
This part is local and we cannot answer it for you. Some places take a fully spent steel aerosol in the household metal stream. Some want it at a collection point along with paint tins and solvents. Some want the plastic cap and trigger off first, and some do not care.
The reliable move is to look up your own council or municipality once, write the answer somewhere, and stop guessing. It takes five minutes and it covers every can you will ever finish.
Keep the straw. It fits the next can, it is the piece that goes missing, and there is no reason for it to leave with the tin.
Keep the cap too if the next can comes without one, which happens in the bigger multipacks. Neither part is worth much on its own and both are annoying to be without.
Dust-Off