Well, what's his work relationship with this woman? Is she his boss? I ask because any time the holidays come, a birthday comes, or I go out of town and my bosses (both married males), are aware of the fact I am going out of town to another country, I make it a point to bring them something small (less than $10), a little souvenir, out of consideration, and because of the fact they are my bosses and I have left them stranded without me being able to perform my job, so it sort of makes up for it. If she is his boss, then I think it's okay that he brings her something small when he is out of town. However, if they are at the same level of the corporate ladder and he only brings gifts for HER, and not the other co-workers, I'd also be thinking it's wrong. Anytime I bring a close co-worker a gift, I make it a point to bring the other co-workers something smaller/cheaper, but SOMETHING, so they don't feel left out, as I find it rude to give someone a gift in front of others and excluding the others, unless it's a boss, in which case, you're allowed to do this, but not a fellow co-worker.
Also, does he go to lunch with this woman alone all the time? I have gone out with my male bosses rarely, but only to discuss business, and I can count the times I have done so on one hand. I have no interest in them and I am sure that feeling is mutual. On occasion, another co-worker may join in, but if your husband is going out alone with her, it may look suspicious to others, and apparently it has, as another co-worker is trying to accuse him of hooking up with this woman.
For a co-worker to make such a strong accusation, there must be reasonable suspicion; maybe winks, giggles, patting on the shoulder, texting... just because you haven't seen the texts, doesn't mean they AREN'T there, he could just be deleting them. As others at his workplace are noticing, there must be something going on that is work-inappropriate and has crossed the line. He ought to be careful, as he could get reported and lose his job if anyone senses some sort of attraction between the two as most companies do not allow fraternization. This happened to a single male friend of mine, who was spending too much time at his single secretary's desk, and someone decided to report him for trying to hit on the secretary and he lost his job.
As to your husband needing her to download songs on the mp3 player, come on, seriously. If she can download songs, there's no reason he can't. Is he computer illiterate? He has probably decided to give the mp3 player to that woman and is using the download excuse.
If he is interested in mentoring her as he says he sees her as a daughter, I don't see why he didn't come to you and say "hey honey, there's this young girl at work who I think we should mentor for x reasons, why don't we set up lunch next weekend and you can meet her?" For him to never want to introduce her to you or do a double date with her and her husband seems like he is hiding something, from both you and the woman's husband.
I don't see why he can't be more open with you if he sees how much this is hurting and upsetting you. If he cares about you, he would stop hurting your feelings and keeping his friendship with this woman strictly cordial and work-related.