Where happens MHCII coordination to a self-peptide, and what happens if the bond would be irreversible?
It appears that in cases self-pepties are bound to the MHC-II proteins (say HLA-DP2) when they are at the surface already. Where do such self-peptides come from?


Class II MHC molecules are capable of presenting any peptide of appropriate length, regardless of where it came from. Contents in a phagolysosome will contain both self and non-self peptides.


  Do they come from inside the endosomal active cell


Some do, yes. While many proteins are degraded by the proteosome, some are actively imported into a lysosome and degraded there. This is termed chaperone-mediated autophagy. Full organelles may also be brought into the lysosome through "regular" autophagy.


  or could they also come from "outside" from where the T-cell comes?


This is the case as well. Any ingested material from the outside could contain cellular debris, protein agglutinations, or foreign organisms. Peptides from all of these sources can be presented at the cell surface.


  What would happen if (for some reason) a) the self-peptide binds "irreversibly" to the MHC-II?


According to this study, MHC II molecules are recycled, and their turnover is carefully regulated. There is no need for the bound peptide to dissociate during antigen presentation, so likely problems would occur when MHC II turnover is disrupted.


  the self peptide binds irreversibly to the MHC-II and at the same time is structurally significantly distorted (e.g. if there is something, like a heavy metal bound in-between the MHC-II and the self-peptide)?


If the MHC itself is structurally distorted, it will not be recognized by any other cells. Remember that, during T cell selection, the cells must be able to recognize the MHC itself with a low affinity. T cells which bind strongly to the MHC itself undergo death by negative selection, whereas those which do not recognize the MHC at all undergo death by neglect. As a significant structural change would result in either an excessive affinity, or no affinity, no T cells capable of recognizing the complex would be present.