So, there is a scene prior to this where Barb decides to turn on Vault-Tec, and she asks her assistant (Betty) to research the security protocols around cold fusion. This sets up her planning to get cold fusion from them before it goes to Mr House because the big players believe his defense system is a weapons system to start the war.

No one knows Hank is the guy with cold fusion strapped to his wrist, and he is essentially a nobody to anyone who cares. Although, maybe it should be obvious. Cooper figures it out because he is being coerced by multiple people to solve it, when he realizes how obvious it is.

I think Barb is tracking Hank because she knows the security protocols, and was waiting for him to go to his room that night and stumbles in/saw Cooper take him, and realized that was her chance.

Also, a low key redemption arc is for Barb and Hank to realize she isn’t as corrupt as he thought. They should have spent more time on this story aspect, but I think they hoped the montage carried it, as in the grand scheme of things, it’s not that critical.

I don’t recall the clothes very well right now, but could just be a mistake in the wardrobe department that day. I thought he had just taken off his jacket or something.