"He is not doing well."
Translation:Li ne fartas bone.
I wouldn't say so. Not doing well isn't the same as doing bad. It implies it, but to a lesser extent just doing outright bad. Also, taken literally, not doing well could simply mean you're doing okay or are content, rather than necessarily negative. However, in English, it generally implies that you aren't content - still, to a lesser extend than bad.
Those are mostly connotations though, I'm not sure how well those translate to Esperanto.