A "rule" pretty strictly adhered to in formal diction, and mostly adhered to in speech, is that we do not employ the present perfect with time adverbials which do not include the present. Under this rule, b (He has gone out five minutes ago) is excluded, because any point in time designated with the preposition ago, no matter how recent, excludes the present.
- However, if the time adverbial is marked as a "supplement", something not integrated into the main clause but tacked on as an afterthought, the sentence would be acceptable: He has gone out—five minutes ago.
c, He had gone out five minutes ago, is also excluded, because ago in ordinary speech establishes a point in time relative to 'Speech Time', the time at which the sentence is spoken; but the past perfect locates an event as past relative to a different time which you are talking about, your 'Reference Time'.
However, this sentence would be acceptable in certain literary uses. Virginia Woolf, for instance, was very fond of reporting her characters' thoughts and words with past tenses, in the same timeframe as her narrative, but leaving the incidental adverbials in a present timeframe to preserve immediacy and colloquiality:
It was terribly dangerous work for a one-armed man, she exclaimed, to stand on top of a ladder like that — his left arm had been cut off in a reaping machine two years ago.
That leaves only a, He went out five minutes ago, which is the natural way of expressing this thought.