You're rheumatologist is right - it could take that long. It could take even longer, though. Many don't see the full benefit of plaquenil for as long as six months. It does seem to work well for fatigue and joint pain. It sure kicked the fatigue for me. There are side affects for some, but for most it's pretty easy to tolerate. The burning and numbness could be some kind of neuropathy - but it could be a number of other things, too. You might need a neurologist to check that out.
Hair loss isn't a 'given' - I have had a little thinning, but really nothing I would qualify as 'hair loss.' Everyone is different.
"If you trust Google more than you trust your doctor than maybe it's time to switch doctors."