If I wear my Class A uniform to a wedding, do I also wear awards on my shirt (in case I take off my jacket)? What if I'm wearing a bow tie?

I don't know about today's regs, but in my time if you wore the blues with a bow tie, It was a formal uniform and you therefore never had the option to wear it without the jacket at all. You had to leave it on. But here's my wedding story from a million years ago that shows how sometimes you just go with the flow.

My bride's parents were from Latvia, and the ceremony/reception was at a Latvian church in the states. Most of the guests were Latvians. So after we eat at the reception, my new brother in law comes up to me and says "dad wants to know when you are going to take off your coat". I told him we dont take them off in the Army. He went away, and then came back and said to my wife "dad says for you to tell Mike to take off his coat". Perplexed, I stood up and took of my coat. All of a sudden, we had the Latvian guests up dancing on tables and generally going crazy. Turned out they were all waiting for the groom to signal "it's party time!"by taking off his coat.

So as another poster said, feel free to make adjustments to formal regs as needed. I seem to vaguely recall a bunch of officers taking off their coats during a dining in before we engaged in playing broom hockey in the Ft Bragg OClub ballroom. And our S3 definitely had his coat off with bow tie on when he broke his leg seeing how many dinner chairs he could leap over.

It is really simple - If you are wearing a "Dress" uniform and take off your jacket (blouse), you are not wearing a Class B uniform; so do not wear anything on your shirt.
Consider this uniform as a Black tie uniform as stated in the Pamphlet, and the equivalent of a civilian tuxedo.

No - usually if you are wearing the As coat, your stuff is there. If during the wedding you take off your jacket, you should heave the long sleeve white shirt underneath, without name tag or ribbons. "Class Bs" is a variation where no jacket is being worn, and usually short sleeve shirt.