It is hard to tell what is going on, but it sounds like she's just wanting someone to talk to and help her with favors (like getting her car serviced). Perhaps she has no family or friends/co-workers she can rely on for things like this. It doesn't seem like he is spending time outside of the home with her. He is picking her up at the dealership and taking her to work because it is convenient, and cheaper than her calling a cab back and forth. Whenever he has checked on her apartment, it's when she has been out of town, so I don't think anything unusual is going on.
We have a female employee here who checks on her married male boss's apartment, waters plants, picks him up from the airport in a rental, etc. She is his assistant and gets paid to do whatever her boss wants her to do. She has been there for movers, a/c repair man, etc. If your husband and this woman were going to happy hour and things like that alone, together, or he is cutting down on your weekend time to be with her, perhaps then I would worry something is off, but their interactions seem innocent. I would just keep the communication with him going, and perhaps suggest dinner whenever he is planning to do favors for her, just to see if you pick up on some odd vibe when they are together, but I don't think anything funky is going on here.
I am separated and have many male friends, some of which are married and I have NEVER had any affairs with them, nor has the thought crossed my mind. We talk about our kids, school, work...mundane stuff. Married women can have affairs too, I don't think it is fair to assume all women who are separated or divorced are hunters looking for prey. When he says he will keep a friendship with her, it may just mean he is willing to keep in touch, it doesn't mean he is romantically or sexually interested. In the business world, especially if she is a boss and has some degree of influence (him getting a raise, and assisting him in getting this job), the last thing you want to do is burn your bridges, so his friendship with her may also be for networking purposes. Wait and see how their friendship develops now that he is working elsewhere. I have a feeling it will cool off.
PS: The fact he is texting and talking to her while he's home rather than secretly doing this (and him telling you about their conversations, etc.) is a good sign. I think if he were hiding his phone, putting it on mute, and pretending he isn't talking to her then that would be when you should start worrying. Seems he has nothing to hide, someone who is up to no good will be hiding things and their conscience will make them act nervous and uneasy.