Showing the decrypted sum of encrypted values Is there a system that would allow to encrypt values with one or more keys, sum the encrypted values, and reveal a key which could only decrypt the sum.

Essentially would be able to show encrypted values where anyone could verify the sum by adding all of the encrypted values, then decrypting it.
There is a recent line of work that does exactly this; it is called functional encryption for inner-product. It allows to encrypt a vector of integers (from some exponent space $\mathbb{Z}_p$) so that one can generate secret keys that do only allow to decrypt a given inner product between the components of any such encrypted vector.

There is a simple construction under DDH, and under LWE. The construction from the Paillier cryptosysem is way more involved and was presented at the CRYPTO conference in august this year.

For the DDH-based variant, the intuition is simple: encrypt your vector with different keys, but the same random coin each time (id est: $E((m_1, \cdots, m_n);r) = (g^r, h^r_1g^{m_1}, \cdots, h^r_ng^{m_n})$, where $g$ is a generator of some group where DDH is hard, and each $h_i$ is $g^{s_i}$ for some secret key $s_i$). The secret key to decrypt an inner product with $a_1, \cdots, a_n$ is just $\sum_i s_i a_i$.

Here is the original article: http://eprint.iacr.org/2015/017. If you want to look into the more advanced constructions, I suggest looking the papers that cite this one.

EDIT: I forgot to mention, the DDH-based instantiation has the same drawback that the additive variant of ElGamal it relies on, which is that a discrete logarithm must be performed on $g^{\sum_i a_i m_i}$ at the end, so it works only when $\sum_i a_im_i$ remains small (say, less than $2^{30}$). If the messages can be large, then the LWE-based or the Paillier-based instantiations must be used.