It is believed that Lyme can trigger the onset of autoimmune disease. In fact, this is one of the "pet" theories put forth by the CDC to explain symptoms after what they recommend as "adequate" Lyme treatment. Unfortunately, they don't have proof that there isn't an ongoing lyme infection at the root of this supposed autoimmune process...
So to answer your other questions, we don't know for sure. Some feel they have sufficiently conquered the Lyme and are only dealing with damage caused by the Lyme infection. Others feel that they have a concurrent infection plus autoimmunity that was triggered by the Lyme.
And it is possible that someone could have an autoimmune condition already, and then become infected with Lyme, so that person would most certainly have both. I know there is a chicken-and-egg argument here, but I do not believe all autoimmune disease is Lyme even though I do believe Lyme is fully capable of triggering autoimmunity.
Can the other disease go away once the Lyme is gone? Possibly, but not definately. I have heard of others finding relief from various diseases after getting treatment for Lyme...and I have heard of some who still need to deal with their other disease(s) after Lyme treatment. It depends on the individual/situation.
The above is my opinion only. I am not a doctor, I just read a lot.
Chronic Lyme Disease, Chronic Bartonella (clinical dx only), Gluten & Sulfite Sensitivity, Many Food/Inhalant/Medication/Chemical Allergies & Intolerances, Asthma, Gut issues (dysmotility, non-specific inflammation), UCTD ("Secondary Lupus-Like Syndrome"), Osteoporosis, chronic Lymphopenia, etc.; G-Tube; Currently weaning off TPN.
Meds: Pulmicort, IV Doxycycline, Heparin (to flush PICC line), Claritin, Singulair, Domperidone, Colloidal Silver (topically & nasally), probiotics, Liver support herbs, Ailanthus, digestive enzymes, homeopathy.