Some men from the south are a little different than anywhere else some are more tentative to their women and their needs and making them feel special. I wouldn't worry too much if your daughter feels uncomfortable then I would say something to her but I'm sure she can handle herself and he very well could be falling hard for her. Just tell your daughter you are worried and ask her not to rush anything, she knows what she can handle and what she cannot. If she is pretty and has a great personality to go with that then why wouldn't any man wants someone like that in their life and have the best of both worlds. A week is not a long time to know someone and it could be that he is using all the time to get to know her and see if she is the one, many couples during the first week are trying to spend time to see where things can go. Now if he told her he loves her during that first week then I would be a little worried but alot of men from the south look at relationship much different than men from big cities. All you can do is tell her to be careful and take things slow.
That is the sign of a good mother, one that worries about their daughter even when they are grown up and out of the house but as far as this guy is concerned he may have a new lease on life and doesn't sweat the small stuff and he may like a woman with meat on her bones. since he is doing all of the pursuing he was attracted to her and pursued her. Yes you're right asking the boss out may not be the norm but if he was attracted to her he probably didn't want to let the opportunity go and possibly lose her to someone else. It's important that they keep their relationship outside of work so not to get into trouble. Since he is paying for everything I doubt he is using her now if he asks her to help him advance then I would put my foot down if I were your daughter, just tell her to look for warning signs that don't sit well with her like, his behavior changing, getting too aggressive, not paying for things anymore, touchy feely at work, asking for things he didn't ask for before. If those thing happen I would suggest to your daughter to reconsider that relationship and be careful not to be alone with him at work because if things get bitter she doesn't want him to file a sexual harassment case against her.
What she is going to have do is learn through his actions whether she can trust him or not, trust is earned and not given so she has to take that approach. If you are afraid she will sabotage the relationship because of trust issues then what she can do is talk to him about her past and what she went through in relationships and ask him if he isn't serious about dating her then don't hesitate to tell her before she gets her feelings involved. She can tell him all she expects is honesty and respect that is all she can ask for. Relationship are all about taking chances and she can be cautious but trusting just watch for change.