I live in a state in which my car does not have to pass smog so technically I could have them taken off. Should I do this? How noisy will my car be without them? Will my check engine light turn off?
Definately take them off. Straight pipe it - it's extremely cheap, will increase your power and mileage, and since you don't have emissions testing no one will be the wiser. The added noise is negligible.
The check engine light will stay on but you'll know why and you can ignore it.