On the status of learnability inside $\mathsf{TC}^0$ I'm trying to understand the complexity of functions expressible via threshold gates and this led me to $\mathsf{TC}^0$. In particular, I'm interested what's currently known about learning inside $\mathsf{TC}^0$, since I'm not an expert in the area. 

What I've discovered thus far is:


All of  $\mathsf{AC}^0$ can be learned in quasipolynomial time under the uniform distribution via Linial-Mansour-Nisan. 
Their paper also points out that existence of a pseudorandom function generator prevents learning, and this, coupled with a later result of Naor-Reingold that $\mathsf{TC}^0$ admits PRFGs, suggests that $\mathsf{TC}^0$ represents the limits of learnability (at least in a PAC-sense)
There's a 2002 paper by Jackson/Klivans/Servedio that can learn a fragment of $\mathsf{TC}^0$ (with at most polylogarithmic majority gates). 


I've done the usual google scholaring, but am hoping the collective wisdom of cstheory might have a quicker answer: 


  Is what I described the state of the art for our understanding of the
  complexity of learning (in terms of which classes sandwich efficient
  learners) ? And is there a good survey/reference that maps out the current state of the landscape ?
The main thing missing from your list is the beautiful 2006 paper of Klivans and Sherstov.  They show there that PAC-learning even depth-2 threshold circuits is as hard as solving the approximate shortest vector problem.