A new class of doubly-generalized LDPC codes over extended code alphabets has been recently proposed in [1]. These codes have been shown to have good asymptotic and minimum distance properties in the range of high code rates, and their structure is solely based on non-binary repetition and parity-check component codes. In this work we go further in the investigation of the new code class, called Repetition-parity-Parity (RPP) in the rest of the paper, by extending the definition of [1] to irregular degree distributions. One performs the asymptotic decoding analysis of the RPP ensemble over the binary erasure channel and formulates a degree distribution optimization problem by means of an approximated EXIT analysis. Some insights on the stability, minimum distance properties and low complexity encoding and decoding of RPP codes have been also developed.