1) Testing for those viruses is indicated when a kitten or for a newly acquired cat with an uncertain history. Because they're both immunosuppressive and potentially life threatening they predispose to secondary infections and neoplasia (FeLV only). A vet must know the viral status of a cat before treating.
2. You're describing feline herpesvirus (FHV-1) which is found in 100% of cats in some populations. It can come out of its carrier state in times of stress and so Timothy's sneezing at this time could be due to FHV-1 rather than a respiratory Mycoplasma.
3. When you posted "...reveal low count, requiring transfusion" I mistakenly thought that he had been transfused. Yes, without a transfusion you're likely to lose him but without viral testing you can't know if it makes sense to transfuse him. FeLV and FIV testing is a simple 10 minute in-house blood test.
4. Testing included a complete blood count and biochemical analysis. It revealed profound regenerative anemia. I also noticed a lymphocytosis (increase in the white blood cell lymphocyte) which usually indicates immune system stimulation as would occur with Mycoplasma, FHV-1, FeLV, FIV and any number of infectious agents and immune-mediated disorders including immune-mediated hemolytic anemia which can be tested for by seeing if the blood autoagglutinates (not now because he's being given a steroid) or he's positive for ANA (anti-nucleic antibody). Because immune-mediated hemolytic anemia is an important consideration, his steroid dose should be high - an immunosuppressive dose.
His prognosis is very poor to grave but can't be determined with the degree of specificity I want without additional testing. Considering your financial constraints and prognosis, I would perform the simple in house viral testing, continuing doxycycline, and making sure that the steroid dose is immunosuppressive. Please continue our conversation if you wish.