When using a preposition that requires the usage of dative, does the dative also apply to "ein paar" in this case.
So is it
Ich werde mit ein paar Kindern wohnen.
Ich werde mit einem paar Kindern wohnen.
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Your second sentence is wrong. To make it orthographically correct, you had to write paar with an uppercase P, since it is a noun in this sentence. But then the sentence gets a weird meaning:
Ich werde mit ein paar Kindern wohnen.
I will live with some children.
Ich werde mit einem Paar Kindern wohnen.
I will live with a pair of children.
Why is this?
Ein paar is a phrase that means some. So, if you use einem instead of ein, you no longer use this phrase.
If you use einem, then it is a singular article (a or an in English), so it refers to one thing. So it can't refer to Kinder which is in plural. It only can refer to Paar, so
Summary: ein does not belong to the head noun, which would regularly appear without article in the indefinite plural anyway: "Ich habe Fragen".
We do see inflection e.g for "kein-e weiter-en Fragen", so we conclude that uninflected ein does not fit and must belong to ein paar instead.
ein paar is never inflected. Nevertheless, we see deine paar with inflected pronomial.
Why that is should be hard to tell. It's good advise for the second-language learner to understand the phrase as a bound lexeme, that is markedly different from the dual idiom, Paar (also cp Päärchen), but only in the odd cases, certain combinations of case and gender like fem./neut. dative. It does not differ in masc. nom. ein [P/p]aar Schuhe. This lack is likely the most signuficant factor for the original confusion.
The orthography is mostly utilitarian: Whitespace improves readibility (contrast dasselbe), or rather writability, because we don't know what else ein should be if not an article; Capitalization or the lack thereof marks the semantic distinction, which can become grammaticalized.
Thus it is treated e.g. by Canoo as Indefinitpronomen und Artikelwort.
It should be interesting to know when Paar and paar dissimilated. It seems however impossible to explain retention of ein regularly.
a) ein does not seem to be inflected for the head Noun, fragen.
b) if ein is the article to paar, that implies it were a noun, but then what's Fragen? Noun+Noun nounphrases are odd to beginn with; attributive nouns or adjectival noun phrases do exist: "England-Fahne", "einhundert Fragen"); But we are not dealing with a compound noun, whereas quantifiers regularly omit the article.
In fact, the head noun can be omitted, so that the qualifier becomes nominal: "Ich hab noch ein paar", equivalent to welche, etwas, mehrere, andere, and I'd venture a guess that it's still not capitalized in this position.
c) skip to point 7. for the essential bit.
"Paar", "paar", and En "pair" are loanwords from Latin, vulgar Latin or French. This should be obvious, because p had shifted to f in the Great Consonat Shift and partially again in the High German Consonant Shift. The root of Latin adj. par is uncertain.
par "equal" is akin to impar "unequal; uneven; odd (numerically); inferior". im- ~ in- had shifted to en-, em- in Frankonian dialect (?) and French in various cases, e.g. employ (< implicare), en "in", enter (Ger ein-treten); cp however interessant. The different vocalism need not concern us, if it's chiefly elided in speech: "Ich hab noch'n paar Fragen". Meanwhile, alternation between m ~ n before plosives is historically precedented in German. In fact, inpar is attested as variant of impar in Latin already.
Compare En "a couple odd questions". I think the evidence is clear at this point. ein paar might well be a reanalysis of impar.
Fr impair, En odd, or Ger schief (math. ungerade) exhibit some negative connotation. Consider further the symbolism ingrained in the weak left versus the strong, morally upright right (pertaining to hands), or the saying mit dem linken Fuß aufstehen. Further, phrases like [X und] ein paar Zerquetschte, Kaputte, Abgebrochene ... pertaining to left-overs may be notable. Contrast rund "circa, around; round", abrunden "math. to round down, floor; to smoothen", aufrunden "to round up, assemble, make ready, finish" (of a group of subjects).
English impair literally means "unsuitable". Though it derives differently, from "Old French empeirier, variant of empirier (“to worsen”), from Vulgar Latin *impēiōrō, from im- + Late Latin pēiōrō (“to make worse”), from peior (“worse”)", it should be ompared nevertheless, if the root is unknown.
I suppose the cluster, so to speak, further concerns parieren, penalty, pain, pay-back, as well as fair, unfair, ungefähr. The later represent quite well what I'm trying to get at. Although, if Paar "couple" refers to connectedness, then ein paar, if stemming from negation, would imply unrelated or loosely connected.
Of course, this doesn't explain the usage of ein paar as a quantifier*. "Paar" like couple and similar are generally thought to derive through semantic widening from an earlier bi-lateral meaning; cf. Pfeiffer:
"... (16. Jh.); ursprünglich ungenauer Gebrauch von Paar"
"mhd. pā̌r ‘zwei von gleicher Beschaffenheit’ (als Adjektiv ‘einem anderen gleich’), Entlehnung (nach der hd. Lautverschiebung) von lat. pār ... ‘Paar’ (in der Verbindung quinquaginta paria bovum ‘fünfzig Paar Ochsen’ mit ahd. giwet ‘Gespann’ glossiert)." [Pfeiffer/DWDS.de: paar]
Luther only used the pair sense in the bible (ca. 1534, 1545), but four of 17 search results are at least ambiguous; the other ones are mostly from the flood story; one is updated in the 2017 edition, originally
sieben / vnd ein wenig Fischlin., now
Sieben, und ein paar Fische. (Mattheus 15:34)
The relevant results read
hatte einen Knaben vnd ein par Esel (Richter 19:3),
vnd sein par Esel beladen / vnd sein Kebsweib mit jm (Ri 19:10; 2017:
hatte ein Paar beladene Esel bei sich); notably, "ein par Esel" is not inflected for accusative, unlike "einen Knaben", and unlike dative
je ein Knauf war unter jedem Paar der sechs Arme (Lutherbibel 2017 - 2. Mose 37:21), or dative
mit einem Paar gesattelter Esel (2017, 2. Sam 16:1; originally
da begegenet jm Ziba der knabe MephiBoseth mit einem par Esel gesattelt), so it appears indistinct from the nominative plural (die Esel), cp.
und um das Opfer darzubringen, wie es gesagt ist im Gesetz des Herrn: »ein Paar Turteltauben oder zwei junge Tauben« (3.Mose 12,6-8). (2017, Lk 2,24; curiously, the search did not separately list the line quoted in the quote); it is thus indistinct from the indefinite plural; somebody unfamiliar with the word might just ignore it and understand Er hatte ... Esel bei sich; because, further, other uses make the dual explicit, e.g. "Turteltauben" (from context, assuming doves are not polyamorous), or
je ein Paar, Männchen und Weibchen (2007, 1. Mose), or--in my humble opinion the classic example for pair-hood--
ein Paar Schuhe.
Indeed, regional dialect would just as well say "mit einem Paar gesattelte Esel" (if I say so myself), and consequently "mit ein paar Esel[n]" and "mit Esel[n]". Although, "mit Esel" is less usefull, because it appears like singular, vocative, or at least uncountable. Indeed, the indefinite quantifier "ein paar X" behaves very similar to uncountable quantities, e.g. "ein bischen Wasser". Questions about such quantifiers are often the source of confusion (viz dozens of questions on ELU.SE about "few" vs "little" and the like). Also, the bare standing indefinite pronoun appears as proper noun phrase, like "ein wenig", "etwas anderes", etc. Whereas, the article feels natural to omit from adverbial usage, perhaps more naturally if interpreted as hapology, "Ich hab dann['n] paar Fragen".
Also of note, sein Paar Esel (cp. Ri 19:10 above) is inadvertantly correctly inflected for definite neuter singular. Whereas nowadays usage of the indefinite idiom usually inflects the pronoun for the head noun: "seine paar Esel". E.g.:
Christ, Lena, Madam Bäurin/4
... nur möglich, liebe Adele!« Die Schwägerin erwidert achselzuckend: »Kunststück! Deine paar Kröten und dazu die Preise! Was früher eine Gans kostete, das ...
... Teezeugs machen soll? Das bissel Mehl und Butter und die paar Eier brauch ich morgen fürs Mittagessen. Wenn du ... ... näher. »Soviel ich weiß, hat die Mama auch ein paar heiratslustige Angelgoldhechte eingeladen, und nun ...
[> Lena Christ: Werke. München 1972, S. 694-703.: 4]
Den 4. October 1781.
... um etwa einigen […]-seelen aus dem Wege zu gehen, die Dir Deine Semmel, die Du mehr hast, als Sie, beneiden, weil Du nicht gleich ihnen Maultierhandwerk treibst? und wohin willst Du dich flüchten? nimmst du nicht überall Deine paar Semmlein mit, die Du mehr und leichter hast als Andere? ... [gez. C.A., H.i.S. - Carl August, Herzog in Sachsen?]
[Literarischer Nachlaß und Briefwechsel. Hrsg. von Karl August ..., Band 1 von Carl Ludwig von Knebel, 1835; Google Books]
The latter example appears like uncountable, indeed, viz. "deine Semmel". The single example doesn't carry too much weight. Grimm's DWB probably has more. I really do not to find out, now. The Pfeiffer referrence has to suffice.
If there was anything earlier, then it was completely subsumed by "ein par". Supposing anything else than the confusion noted in 7. almost seems like an anachronism. Keep in mind that French had a renesaince in Prussia, at the latest in 1685, after the formation and consequent prossecution of protestant uprise following Luther.
Anything earlier than that requires a leap of faith. In that sense, I want to suggest a direct comparison to English a few, ultimately from Proto-Indo-European *peHw- "small, little". It is uncontroversial question that *p had been lost in Germanic, becoming f instead. It has become a bit of a controversial question for me which other ways it could have survived; cf e.g. the etymology of platt and flat, for which I've heard different opinions, e.g. Frankish as a source, which is however principally Germanic itself, but with Latin influence; On the other hand, cp apple, Low German Appel vs High German Apfel, thought to be ultimately from PIE *bh, *Hebhol-, though fall, *pel-, Abfall, Fallobst, compares well--might be pure Zufall though. Perhaps cp *h2po- "from", of, too. Overall, direct inheritance of p is summarily excluded by common concensus about the totallity of Grimm's law changing PIE *p to Proto-Germanic *f long before the 4th century AD, possibly before 500 BC. p seems to be however retained in the cluster sp (cp spärlich), and s-mobile e.g. is yet poorly understood. I'm not aware of possible non-Germanic sources for a lexeme *par "(a) few", and there's no evidence of earlier occurance than the 16th century anyhow, which is however no surprise, since book publishing only really took off around that time. Also, both usages would be nigh impossible to tell apart without the modern capitalization.
The existence of similar patterns would be a good indicator of a long standing paradigm, but a single parallel in English, i.e. a couple, might just be a parallel development; all the more if a pair is always dual.
Conclusion: ein does not belong to Fragen. Why that is should be hard to tell. The truth might be ugly, and understood differently by various speakers. Perhaps paar is a bound morpheme lending plurality to the indefinite article ein, which is sorely needed because no plural indefinite exists in the articles paradigm; like the definite plural article it does not inflect for acc. Or it's a present participle from a strong verb used adverbially, at least in "deinen paar". Or it's actually a noun that is irregularly not capitalized. One simply cannot hear any difference.