I'm not sure that your problem is entirely an issue of relations and foreign keys.
One issue I see is the error "Update requires a valid DeleteCommand when passed DataRow collection with deleted rows."
Was this code generated by the designer or did you write it by hand?
This sounds like the error you get if your tableadapter does not have a valid DeleteCommand property (you can also geta similarerror for a missing UpdateCommand). Open your MyTableAdapter.designer.cs code and look for a "Delete(...)" method. If it isn't there, I'm betting the datatable involved doesn't have a primary key, or at least a unique constraint in which case the designer won't create the DeleteCommand, UpdateCommand, and the associated methods for you, since you can't update or delete without a unique identifier.
I'm guessing it's the child table that is the problem.
hth,
dob