I think it is a coincidence, though. Your wife's rash sounds very non-specific, meaning that it could be a mild virus in the system or something like that. Eczema sounds much more likely than a yeast infection, by the way. I suggest that she see a skin doctor as soon as possible, so you'll both know what it is, and she can get treatment if she needs it. As to your guilt and anxiety, I'll leave that to you to work out. But I do not think you have given your wife (or yourself) AIDS.
Good luck.
Dr. Rockoff