The integration of multicarrier modulation and multiple-input-multiple-output (MIMO) is critical for reliable transmission of wireless signals in complex environments, which significantly improve spectrum efficiency. Existing studies have shown that popular orthogonal time frequency space (OTFS) and affine frequency division multiplexing (AFDM) offer significant advantages over orthogonal frequency division multiplexing (OFDM) in uncoded doubly selective channels. However, it remains uncertain whether these benefits extend to coded systems. Meanwhile, the information-theoretic limit analysis of coded MIMO multicarrier systems and the corresponding low-complexity receiver design remain unclear. To overcome these challenges, this paper proposes a multi-slot cross-domain memory approximate message passing (MS-CD-MAMP) receiver as well as develops its information-theoretic (i.e., achievable rate) limit and optimal coding principle for MIMO-multicarrier modulation (e.g., OFDM, OTFS, and AFDM) systems. The proposed MS-CD-MAMP receiver can exploit not only the time domain channel sparsity for low complexity but also the corresponding symbol domain constellation constraints for performance enhancement. Meanwhile, limited by the high-dimensional complex state evolution (SE), a simplified single-input single-output variational SE is proposed to derive the achievable rate of MS-CD-MAMP and the optimal coding principle with the goal of maximizing the achievable rate. Numerical results show that coded MIMO-OFDM/OTFS/AFDM with MS-CD-MAMP achieve the same maximum achievable rate in doubly selective channels, whose finite-length performance with practical optimized low-density parity-check (LDPC) codes is only 0.5 $\sim$ 1.8 dB away from the associated theoretical limit, and has 0.8 $\sim$ 4.4 dB gain over the well-designed point-to-point LDPC codes.