Prednisone usually lowers White blood cell counts. So fever + high white blood cells = your immune system is fighting something (lyme, viruses, etc. or autoimmune, but I think there are enough infectious organisms floating around to keep the immune system quite busy).
Heparin won't dissolve existing blood clots, it only prevents the development of more clots, so it may not increase platelet counts. Liver issues can decrease platelet counts also, but I would ask your doctor where you are getting Lyme treatment if they think the low platelets are from Lyme/coinfections or Hep C or if they know?
High red blood cells could just be from dehydration...and also the body will generate more red blood cells when one is at a higher altitude for a long time, but I don't know how long a person has to be at a high altitude before the body starts to make more red blood cells. There are other things also that can cause this but I can't remember what they are off the top of my head...for a list, see http://www.labtestsonline.org/understanding/analytes/rbc/test.html
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.