Results on universal approximation for learners other than ANNs
Let me attempt to partly answer the learning part, or to at least address some connections.
In learning (distributional issues aside), we usually want our algorithms to be able to not only approximate the target function (from a class) with a hypothesis, but also to be able find such a hypothesis (efficiently).  So, even though a certain class can universally approximate a given target, we are not done.

I am not an expert in neural networks, but training a neural network is not easy.  We have algorithms for training a neural network, like backpropogation, but they won't always converge, or converge quickly enough, to meet the PAC learning criteria.
Boosting, on the other hand says (roughly) how to turn a weak "approximator" into a strong one, but it in some sense assumes the difficulty away but giving the learner access to the weak approximator.