In this work, we study two problems: three-user Multiple-Access Channel (MAC) with correlated sources, and MAC with Feedback (MAC-FB) with independent messages. For the first problem, we identify a structure in the joint probability distribution of discrete memoryless sources, and define a new common information called ``conferencing common information". We develop a multi-user joint-source channel coding methodology based on structured mappings to encode this common information efficiently and to transmit it over a MAC. We derive a new set of sufficient conditions for this coding strategy using single-letter information quantities for arbitrary sources and channel distributions. Next, we make a fundamental connection between this problem and the problem of communication of independent messages over three-user MAC-FB. In the latter problem, although the messages are independent to begin with, they become progressively correlated given the channel output feedback. Subsequent communication can be modeled as transmission of correlated sources over MAC. Exploiting this connection, we develop a new coding scheme for the problem. We characterize its performance using single-letter information quantities, and derive an inner bound to the capacity region. For both problems, we provide a set of examples where these rate regions are shown to be optimal. Moreover, we analytically prove that this performance is not achievable using random unstructured random mappings/codes.