In the meanings you cite, jimmy is from 1848, jack is American English from 1904, peter out is miners' slang from 1846, and the two johns are from 1932 and 1911, respectively.
The only connection I was able to find between any two of them is this:
john "toilet," 1932, probably from jack, jakes, used for "toilet" since 16c. (see jack).
As to the aside, yes, it does happen in other languages, too. As the most obvious example off the top of my head, the Russian derogative term for a German is фриц (Fritz), and the German derogative term for a Russian is Iwan (Ivan).
Edit: seanyboy's answer made me revisit peter out, and it is indeed not clear whether it is connected to the name Peter at all. Etymonline says:
peter (v.) "cease, stop," 1812, of uncertain origin. To peter out "become exhausted," is 1846 as miners' slang.
Wiktionary adds:
Various speculative etymologies have been suggested. One suggestion is that it comes from peter being an abbreviation of saltpeter, the key ingredient in gunpowder – when a mine was exhausted, it was “petered”. Other derivations are from St. Peter (from sense of “rock”), or French péter (“to fart”).
Alas, both words most probably do not come from the same root. So far as we know, their identical spelling is completely accidental. They were already spelled the same more than two thousand years ago by the Romans.
Our word German comes from Latin Germanus, first attested in Caesar, which was used to describe the Germanic tribes by the Romans. According to the Oxford English Dictionary (2nd ed.), it may be of Celtic origin, as a name used by the Celts to designate their neighbours; but there appears to be no consensus yet.
The word germane comes from Latin germanus (adjective "related, relative", noun "brother") This comes from the Proto-Indo-European root *gen-, which means something like "give birth". Our word germ, from Latin germen, "sprout", is from the same root, as is genetic, through Greek genesis, "birth".
Best Answer
If you look more closely you will see that the hi- was there originally, in Greek historia from which it was borrowed into Latin. The initial syllable was weakened and sometimes dropped in Late Latin, and reduced to e- in Old French, from which the word was borrowed into Middle English.
In ME it shows up as historie, istorie, estorie and histoire, all representing OF forms, probably influenced by Latin – for of course Latin was still a living written language of learning and scholarship. Alongside these a “native” version, with the initial syllable entirely dropped, began to show up; this appears as storie, stor, storri, with plurals stories, storise, storius, and storien.
All these forms were used indifferently for any narrative account, whether formal chronicle or patent romance. It was not until Early Modern English – the 16th century – that spellings and forms began to shake down– probably, again, influenced by the status of Latin as the principal language of learning – into the contrasting history = factual narrative and story = fictional narrative.
Note that to this day French histoire means both story and history – as does the corresponding term in German, Geschichte. I imagine this is true in many other European languages.
EDIT:
This simplified contrast is rightly challenged by Arlen Beiler and John Lawler: story embraces any narrative, not only fictional narratives, and the two terms have never completely separated. But by and large, history has come to mean the product of the academic discipline, while story has come to mean an engrossing narrative. OED 1 puts it rather neatly, I think, under Story 4 e [story of a life, institution, etc. ]:
I must also acknowledge that over the past two generations historiographers have grown skeptical of the Rankean eigentlich gewesen and are much more conscious of the element of mythopoesis in their work; so in a sense history is collapsing back into story.