We address a design of high-capacity and low-peak-to-average power ratio (PAPR) orthogonal frequency-division multiplexing (OFDM) systems based on bit-interleaved coded modulation (BICM) utilizing non-equiprobable and non-uniform (NENU) constellations as well as clipping and filtering (CAF). The proposed constellations are generated using a truncated Gaussian distribution and the merging of constellation points, where the former creates a non-uniform constellation (NUC), and the latter adjusts the number of signal points for further improving the total bit-wise mutual information (BMI). Unlike other exhaustive search-based approaches, the proposed constellations are uniquely determined by only two parameters associated with NUC and cardinality. Due to this property of limited degrees of freedom, the complexity required for the numerical optimization process can be significantly low. We focus on the constellation design based on one dimension, i.e., pulse amplitude modulation (PAM), which facilitates the reduction of demapping complexity for the BICM receiver. The use of CAF at the transmitter can efficiently reduce the PAPR of OFDM signals; however, it introduces clipping noise that may degrade error rate performance, making the application of clipping noise cancellation (CNC) at the receiver essential. Therefore, we optimize the NENU constellations in the presence of CAF and CNC. Simulation results demonstrate that the combination of constellation shaping with CAF and CNC enables BICM-OFDM systems to simultaneously achieve low PAPR and high spectral efficiency over additive white Gaussian noise (AWGN) as well as frequency-selective fading channels. Furthermore, comparative studies confirm that the proposed system significantly outperforms the single-carrier counterpart (i.e., DFT-precoded BICM-OFDM) in terms of PAPR and bit error rate (BER) performance over fading channels.