Smartphone camera wired trigger over audio
Yes, you can. 

Yes, you can omit them. Look at handsfree headsets. Same exact thing as headphones except only one speaker.

And maybe. The phone may not recognize a remote is connected if the mic is completely missing. You are essentially making what's known as a quick klick (click) button. Simple and done by many, as well as super cheap commercial versions.

Keep in mind the Wired Audio Headset Specification is for devices to conform to Android Specs. It is required that android devices support the feature, not a requirement that the headset have the feature. Even then, it's just "strongly recommended" that the device support it.