When Mac Taylor and Stella are talking about the cigarette experiment, as Stella leaves, you can clearly see that the cigarette is firmly behind Mac's ears, but then the scene cuts to just Mac alone and the cigarette is barely on top of his ear, making it easier for it to fall off Mac's ear.
The cigarette butt found in the ground simply wouldn't have existed. It takes, on average, five years for a butt to biodegrade, and a maximum of about 12 years. As this was under a football pitch, that is watered very often, five years would be most likely.