What are your actual lab results - TSH, Free T3 and Free T4? Without knowing the actual results for your tests, along with the lab's reference ranges, it's very difficult to comment constructively on your question.
I do agree with your doctor about self-medicating -- never a good idea. Secondly, I doubt seriously that product you purchased from the internet was true synthroid, because synthroid does not contain T3.
Again, without your actual lab results, including Free T3, I can't comment on the need for T3 med.
I couldn't say much about the T3, but, about the eczema and knuckle swelling, a lot of thyroid problems are rooted in an underlying auto-immune disorder.
I used to get pretty bad eczema on my arms when I was younger, though I never had my thyroid levels checked during this.
I'd be concerned about the knuckle swelling. If you haven't already, you might want to see a rheumatologist.