The rash can pop up randomly anywhere on the body. And you're very right, the only thing that produces that bulls eye rash is Lyme and they need treatment.
Yes, Doxy is the go to abx for the first line treatment, but may not be a good choice if she's had the infection for very long as it can encourage the bacteria to go into hiding. At two weeks, she may be right at the 'edge' of it working well still. From what I understand, four weeks is basically too late for many people. The real key is her symptoms. If she starts to have body symptoms, then the infection has spread throughout the body and she will need the "full treatment" not just a couple months of Doxy.
The dose is 300 - 400 mgs daily for 6 - 8 weeks, unless symptoms show up.
www.ilads.org/lyme/treatment-guideline.php