Using SHA3 as an entropy pool Has any work been done on analyzing the SHA3 sponge construction for use as an entropy pool (e.g. /dev/random)?

All the use cases I've heard of consts of a sequence of input operations followed by a sequence of output operations where as an entropy pool would require security to be maintained for any sequence of intermingled input and output operations.
As Paŭlo Ebermann already mentioned in his comments, SHA3 can indeed be used as a pseudo-random number generator.

The paper "Sponge-based pseudo-random number generators" talks about just that and it also describes a clean and efficient way to construct a re-seedable PRNG with a (Keccak) sponge function. What you'll get is a PRNG based on a cryptographic hash function… with the usual security implications. 

For example: the paper explicitly states that you should reseed regularly with sufficient entropy to prevent an attacker from going backwards on the period of the PRNG (which is probably what you've been hearing about).

Related to proof of security of such a sponge-based PRNG construction, the paper states that the presented PRNG has the property of indifferentiability (which practically claims it is indifferent to an ideal PRNG), a resistance against state recovery (which is a result of it's indifferentiability), and that it passed the statistical tests proposed by NIST. This is the result of the fact that the sponge-based PRNG construction inherits the provable security properties of the sponge construction itself. 

I would like to advise you to check on the paper yourself to grasp all the details; like learning how and when to re-seed the PRNG to prevent some potential attacks. It answers your question with ample detail.