My girl threw up once because I scolded her, they just get stressed and upset very easily. Sleeping with mommy will do the trick
at night.
Regarding the barking, give him a treat to keep him busy while you rush to the door. Once he sees there's no one to hear him he will stop. If he throws up and has diarrhea while your gone he probably has separation issues (anxiety).
Don't bath him or take him to the groomer until he feels/knows his at home. Might be to overwhelming for a lil' fellow.
He is soooo cute, Congratulations and Welcome!!