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One thing I've noticed about the usage of ending a sentence with a preposition is how similar the construction is to German separable verbs. With German separable verbs, the prefix is often a preposition when taken by itself, such as "mit-kommen" = accompany, but is "come with" if it's broken into parts. In a regular declarative sentence in German, this would be written with the prefix at the end, such as "Paul kommt Erich mit" = "Paul accompanies Erick."

My question is, aside from the parallel, is there an actual causal relationship here? Is the common usage of putting a preposition at the end come from an old German construction with separable verbs?

This won't be an answer, exactly, but more of a pointer towards other sources.

First of all, terminology. What you are talking about is called preposition stranding. It seems to be very rare (although not nonexistent) outside Germanic languages. How exactly to analyze it within Germanic languages, including whether there is a unified treatment, remains an open problem. And I will stop there, and just give a bunch of sources.

On the Linguistics Stack Exchange: What motivates/allows preposition stranding in English, but disallows it in other languages, like Mandarin?

On Wikipedia: Preposition stranding.

And then come papers and such.

This Master's thesis has a good literature review.

This book, The Germanic Languages, discusses the topic of preposition stranding.

English as North Germanic discussess it, too.

On the historical development of preposition stranding in English

Case Theory and Preposition Stranding

Early Germanic preposition stranding revisited

Preposition Stranding and Resumptivity in West Germanic

Preposition-Stranding and Passive

Preposition Stranding, Passivisation, and Extraction from Adjuncts in Germanic

And many more.

Old English (OE) had sentences that ended in prepositions:

gað ge beforon; ic eow cume æfter.
_{Gregory the Great, Dialogues, Book 1}

(Literally: Go ye before; I you come after.)

OE also allowed the order preposition, object, verb:

Þis hé spræc on Iudea-lande: ðær wæs án eowd of ðam mannum þe on God belyfdon on ðam leodscipe.

This he spake in the land of Juda: there was a fold of men who believed in God in that nation.
_{The Homilies of the Anglo-Saxon Church by Ælfric, translated by Benjamin Thorpe}

See also my answer here about believe in vs believe on. One commenter (KarlG) mentioned that this word order is "still strictly observed in modern German, a bit less strict in Dutch".

However, according to The Ban on Preposition Stranding in Old English "the phenomenon of preposition stranding is highly restricted in OE". The paper points out some examples of sentences where Modern English would likely end a sentence with a preposition, but OE wouldn't. For example:

Drihten, þu þe gecure æt fæt on to eardienne

Lord you yourself chose that vessel in to live

‘Lord, you chose that vessel for yourself to live in’
_{The Blickling Homilies}

I don't think that English ever had "German separable verbs".

In particular, this fairly comprehensive summary of phrasal verbs in OE doesn't mention the form you're asking about:

Old English generally did not possess phrasal verbs as they are found in Present-Day English. They did exist, although they were rare. Much more common in Old English was the inseparable-prefix verb, a form in which the particle was attached to the beginning of the verb. These Old English prefixed verbs are directly comparable to current phrasal forms. For example, in Present-Day English, there is the monotransitive verb “to burn” and then the phrasal monotransitive “to burn up.” Old English had “bærnan” (to burn) and “forbærnan” (to burn up). The prefix “for-” remained affixed to the verb and could not move as modern particles can. Such Old English compound verbs were also highly idiomatic, in that the meaning of the compound form did not necessarily reflect the meaning of the root. Denison provides “berædan” as an example because it meant “to dispossess”, while its root verb, “rædan”, meant “to advise”. The phenomenon still survives today in the participle “forlorn”, as well as the verb “understandan”, which does not in Present-Day English mean “to stand underneath (something)”, but idiomatically “to comprehend”. Akimoto suggests that Old English prefixes often remained before the verb because Old English had strong object-before-verb (OV) tendencies, whereas Present-Day English is largely a VO language, which has made it possible for particles to travel to post-verbal positions. Some Old English verbs did function as modern phrasal verbs do.
_{The Historical Rise of the English Phrasal Verb}

In English usage, even in formal English usage, it is by no means 'incorrect' to end a sentence with a preposition. This supposed rule is attributed to John Dryden(1631-1700) who was influenced by the fact that in classical Latin such a thing is not allowed (https://en.wikipedia.org/wiki/John_Dryden).

As (Fraser, Plain Words, HMSO 1973) notes: Sir Winston Churchill "once made [a] marginal comment against a sentence that clumsily avoided a prepositional ending:'This is the sort of English up with which I will not put'". In that sentence 'put up' is a phrasal verb and unlike German trennbare Verben the prepositional 'up' definitely should not go at the end of the sentence.

A fine example of prepositions at the end of a sentence is this exclamation from a man who has just dropped a cufflink inaccessibly behind a cupboard:"Come out from down in under there!"