There may be hunting missions, but PETA would probably get mad. And yes, there are emotes that your guardian can use to interact with others, since voice chat won't be a big feature outside of your fire-team. Also, in reference to your mission question, you are free to do whatever during the time you are engaged in the quest. Bungie has even said that it has put in instances to optionally divert you from your mission, and possibly get some new gear.