Question about the expression of MHC
Sure generally any cell that expresses MHCII will also express MHCI. Most cells in the body express MHCI as a self-identifier and to present intracellular proteins to T cells as a sort of a status update i.e. the cell will constantly be presenting peptides (pieces of endogenous proteins) on its surface bound to MHCI, if the cell becomes infected it could also present viral proteins on its surface with MHCI, which can potentially be recognized by a T cell specific for that particular epitope. B cells and antigen presenting cells such as dendritic cells and macrophages express MHCII but also need to be able to be identified as self through their basal level of MHCI presentation of self-peptides. The two MHC molecules are recognized by different cell types, MHCI by CD8 T cells and MHCII by CD4 T cells. So you could consider them to work independently from one another. The source of antigens (peptides here) and process for presenting them via MHCI and MHCII are somewhat different from one another.