#1 is a nonpredictive (sometimes termed 'assumptive') conditional. It means basically 'On the assumption that he is a qualified doctor,...'. The apodosis in this case (i.e. that he should have been able to diagnose the illness) is presented as an assertion based on something that the speaker, provisionally at least, accepts as true, rather than one whose truth is open to question depending on that of the subordinate clause.
#2 is in essence a variant first conditional, where the protasis refers to a possible present (rather than future) situation and the apodosis replaces standard 'will' with (epistemic) 'should'.
#3 is essentially a second conditional, varying from the standard paradigm only in the minor substitution of 'should' for 'would', though with no substantive semantic difference.
#4 is a variant second conditional, combining a counterfactual present protasis (implying that he IS not a doctor) with a similarly counterfactual past apodosis (he in fact WAS not able to diagnose the illness - i.e. as a result of his not BEING a doctor, either then or now). This combination is not uncommon where a long-term/lifelong identity or characteristic is considered retrospectively as grounds for the fulfilment or otherwise of certain past circumstances.