Sounds like you do have some issues with trust..........which is completely natural. The reality is a lot of people cheat, the saddest fact is they hurt those they love the most for a simple gratification. I wish I had a crystal ball here, and I could tell you wither he is scum of the earth or he is your prince charming.........but hey life doesn't work that way. (As much as we all wished it would).
The reality is there are some questionable details in all this. While phone calls and texts is not a guarantee that there is something going on........the fact that he did not mention it and when it was brought out he acted like it was no big deal would be enough for most anyone to question this.
Him venting to her is not cool....she is your friend and as such he should have known better then to allow it. On the flip side, as your friend she should have told him to talk to you. The reality is he may be perfectly innocent, but some of his actions are questionable at best.
Why would he go to her office after you asked him not to?
Why would he not mention speaking to her this much.....if this was a issue of once in a while it may have slipped his mind. But that must talking should have came out before now........so the question is was he trying to hide it? If so is it because he knew what he was doing was wrong.....or was it because he did not think about it?
Why would he want to go partying with her instead of you? Does he often party with other people? If so then this may not be a big issue.....but the fact is she is your friend and as such this would and should have been something you were invited to.
The happiness could be a simple matter that he likes her as a friend.......the fact is sometimes us guys tend to grasp onto anyone who pays us much attention. If we feel like someone wants to talk to us and pay attention to us, sometimes we will jump on th chance....so this is not a guarantee. The hug thing could be a issue.....depending on his personality. Does he hug other people often? Such as family members, friends, you? If not then this does not look so great on his part.....what kind of hug is it? Does he linger, place his hands in areas that they should not be? If so this is a issue.
At this point you have a couple of issues, one boils down to trust (Which may not be so easy for you). If you trust him then sit down and talk with him. Make a list out of the things that happened and your issues with them. Let him know why they bothered you and how you want him to act in the future. If you want the relationship to end, then be honest. While I do not believe in giving ultimatums, if this is a breaking point for you then you need to be honest and upfront from the get go. Let him know what you want and what you need.
Your other option is to walk away........if you really believe he is cheating then you must do what you feel is right. I can not tell you what to do.....but life is way to short to allow someone to treat you with disrespect. If this marriage is going to work, you both must sit down and talk about this.....let him know that this is a serious issue and one that could end your marriage if he is not honest with you. Let him know what you need and let him know how important this is for you.
As for your friend, I would suggest sitting down with her as well. Let her know how you feel and ask for her honesty in the issue. Let her know that if her man was cheating on her you would women up and tell her, and you would hope in the spirit of all that you have shared that she would be honest with you. If she claims nothing happened explain to her how this all made you feel and ask her to please end the relationship with him if that is what you want. Ask her to cease communicating with your husband. If she is being honest, she will do this. As a women she understands what this feels like so do not underestimate her......if there is no relationship there she will step back as a courteous. If not either she was not being so honest.........or she is one of those vindictive women. Either way if this is the case, better to know now how she is and end the friendship then to go on and find out later.
I know this is hard.........and I wish I had some easy answers for you. The reality is if he is being dishonest with you he will continue to do so. If he is being honest and loves you then ending this friendship should not be a big issue for him.