Does the order in which I turn off the gas stove matter?
I had this type of stovetop for many years. The way I usually did was:


Turn off the stovetop
turn off the canister


The next time you turn the canister on, gas won't leak through the burner you left on (because you didn't!). And the amount of gas left in the connecting hose is negligible and should be safe to leave it there until the next use.