In this study, the performance of generalized low-density parity-check (GLDPC) codes under the a posteriori probability (APP) decoder is analyzed. We explore the concentration, symmetry, and monotonicity properties of GLDPC codes under the APP decoder, extending the applicability of density evolution to GLDPC codes. On the binary memoryless symmetric channels, using the BEC and BI-AWGN channels as two examples, we demonstrate that with an appropriate proportion of generalized constraint (GC) nodes, GLDPC codes can reduce the original gap to capacity compared to their original LDPC counterparts. Additionally, on the BI-AWGN channel, we apply and improve the Gaussian approximation algorithm in the density evolution of GLDPC codes. By adopting Gaussian mixture distributions to approximate the message distributions from variable nodes and Gaussian distributions for those from constraint nodes, the precision of the channel parameter threshold can be significantly enhanced while maintaining a low computational complexity similar to that of Gaussian approximations. Furthermore, we identify a class of subcodes that can greatly simplify the performance analysis and practical decoding of GLDPC codes, which we refer to as message-invariant subcodes. Using the aforementioned techniques, our simulation experiments provide empirical evidence that GLDPC codes, when decoded with the APP decoder and equipped with the right fraction of GC nodes, can achieve substantial performance improvements compared to low-density parity-check (LDPC) codes.