Great question! It seems that this word was originally pronounced with /kw/, as in "quiet," but the /w/ ended up being lost because the following vowel was unstressed.
As Andrew Leach says, in Modern French <qu> generally represents a single consonant /k/, due to a sound change from /kw/ to /k/ that occurred at some point. However, the word conquer was not taken from Modern French: it was taken from "Anglo-French," as you say, and in this dialect of French "qu" does seem to have generally been pronounced /kw/:
words like quit, question, quarter, etc, were pronounced with the
familiar “kw” sound in Anglo-Norman (and, subsequently, English)
rather than the “k” sound of Parisian French. ("Norman Conquest" –
The History of English, Luke Mastin)
The words exchequer, chequer, lacquer, liquor are graphically similar, but they're different from conquer in the specifics of their etymology, so unfortunately a comparison with the pronunciations of these words is not conclusive. According to the Oxford English Dictionary (OED):
chequer and exchequer are ultimately derived from late Latin scaccārium, so the <qu> in these words just reflects the Parisian convention of writing /k/ as <qu>. I can find no evidence that it was ever pronounced as /kw/ in any variety of French or English. In older forms of French, it was apparently spelled with a <k>, and in English, spellings with <k> or <ck> are attested earlier than spellings with <qu>.
lacquer is derived from "obsolete French lacre (17th cent.)." The spelling with <qu> is again due to Parisian spelling conventions; in particular, the OED suggests the French word laque "lac" influenced the spelling.
- liquor does in fact have some etymological basis for the <qu>: it ultimately comes from Latin liquor, related to the root seen in words like liquid. However, the /kw/ in this word seems to have been simplified to /k/ fairly early on (possibly the process was quickened by the roundedness of the following vowel). We find <c> and <k> used in some of the earliest attested English spellings in the OED. A similar case spelled with <gu> seems to be the word "languor", which is ultimately from Latin languor, but which had the following spellings among others in Middle English: langre, langur, langore, langoure.
So essentially, for all of the preceding words, Andrew Leach's answer seems to be the best explanation for why they are spelled with <qu>: it's just a French-influenced spelling of the sound /k/.
For conquer, on the other hand, the "Anglo-French" etymology does suggest the pronunciation should have a /kw/. And there does seem to be some evidence that it was pronounced with /kw/ at the time it entered English. The OED lists some Middle-English spelling variants that use the letter w, cuncweari and conqwere. It also seems like there weren't any spellings in Middle English with just <c> or <k>, which we might expect to see if the word was pronounced without an /w/ in this time period.
As Peter Shor mentions, there don't seem to be any words in present-day English that end in /-kwər/ (or, for that matter, /-wər/ preceded by any other consonant). Janus Bahs Jacquet made the point that this might be considered a phonotactical restriction; I thank him for pointing this out. A Google search turned up the following passage by Julia Schlüter from "Early Modern English: Phonology" (in Historical Linguistics of English: An International Handbook, edited by Alex Bergs and Laurel Brinton):
A minor consonantal change limited to a certain number of lexemes and
dating to Early Modern English is the disappearance of /w/ when
following another consonant and preceding a rounded back vowel, e.g.
in sword, two and who, and somewhat more systematically in
unstressed syllables, e.g. Southwark, conquer, answer. In some
further items, e.g. swollen, swoon, swore, awkward, boatswain,
forward, housewife, and pennyworth, /w/ was later restored on the
basis of the spelling or of related words.
As this passage says, there are a number of exceptions to this sound change.
For example, Peter Shor pointed out that this kind of simplification does not seem to have taken place (or it was reversed) in the related word conquest. The primary stress in conquest, at least in modern pronunciations, is on the first syllable, but the /kw/ comes before a non-reduced vowel /ɛ/ rather than the schwa. Also, there is a similar word quest where the /kw/ does come at the start of a stressed syllable; this may have contributed by analogy to the maintenance or restoration of /kw/ in conquest.
Other words where post-consonantal /w/ before an unstressed syllable was retained, re-introduced, or just plain introduced in unstressed syllables include relinquish (ultimately from classical Latin relinquĕre, via French relinquiss-);
extinguish, distinguish (ultimately from Latin words ending in -tinguĕre);
anguish (from Old French anguisse, angoisse < Latin angustia < anguĕre, a variant of the verb angĕre); languish < Anglo-Norman and Middle French languiss- < ultimately from classical Latin languēre);
vanquish (complicated; the OED says “< Old French vencus past participle and venquis past tense of veintre ( < Latin vincĕre), modern French vaincre to conquer, overcome; the ending was finally assimilated to that of verbs from French stems in -iss-”; to me it looks like the /w/ must also be due to analogy at some point since Latin, since it is not present in any form of Latin vincĕre).
The place names mentioned in the following question may be other examples of this sound change: "The mysterious, unenunciated "w" in the "-wich" of English place names"
I don't think it has to do with them being verbs, and for the most part, I also don't think it has to do with the etymology. I think it mainly has to do with the stress patterns of these words.
The tendency I see is that the letter L is generally doubled after a stressed short vowel, and not doubled after an unstressed vowel. Historically, there has been a lot of variation in the spelling of words like this. Based on etymologies that I have looked up, I would say there is little to no etymological significance to the use of a single L vs. a double L at the ends of words. It's just an oddity of our modern spelling system. (Another relevant question about double L: Why is there a double "ll" in "bell"?)
Here are some actual examples: mill (as you've found out, it comes from Old English mylen), small (comes from Old English smæl), skill (comes from Middle English skele, scele, skile, skyle, from Old Norse skil), dill (from Old English dili, dile, dyle). On the other hand, metal comes from Latin metallum with a double L (although the French spelling metal may have also played a role, as French seems to have been an intermediary in the transmission of this word). I have some more examples of words like metal in my answer here: Adding an L when appending an -ium suffix to a word? (Metallium vs. Metalium)
Monosyllabic content words always end in a stressed syllable, and polysyllabic verbs are more likely than nouns to end in a stressed syllable. These two factors are probably why it seemed to you that this was related to part of speech.
There are many nouns that end in "-ill," especially monosyllables such as bill, hill, sill, and frill. Adjectives such as still can also have this spelling.
There are also verbs that end in "-il." Aside from bedevil, there are cavil, imperil, stencil, fulfil (British spelling; Americans use fulfill). However, it does seem to be true that fewer verbs than nouns end in "-il."
What is the etymology of words that end in -il?
It's worth noting that there aren't that a huge amount of words in total that end in -(consonant)-il."
As you've found, many of them are derived from Latin nouns ending in -illus, -illa or -illum. When Latin nouns are taken into English, final -us, -a and -um are often simply removed, which explains how we get English nouns ending -ill or -il.
Latin verbs, however, are usually borrowed in a more complicated fashion. For complicated historical reasons, Latinate verbs in English are generally derived from the past participle form of the Latin verb. Latin past participles generally have a dental suffix (usually t) that is retained in the derived English verb. So for example, the Latin verb inficio corresponds to the English verb infect. In regular verbs of the first declension (the most common type) this dental suffix is preceded by the vowel a. So Latin verbs containing -ill- have generally been borrowed as English verbs ending in -illate. (For example, scintillate comes from Latin scintillare). That might be a contributing factor to the relative rarity of verbs ending in "-il."
There are various other sources for words ending with "-il." In most cases, the modern spelling with "il" is fairly straightforward based on the etymology.
However, there are a handful of words that seem to have acquired the spelling "-il" for unclear reasons. Here's what I mean: devil (Old English deofol), evil (from Old English yfel), weevil (Old English wifel), stencil (earlier spelled stancel; interestingly, the Oxford English dictionary says that the noun is derived from the verb). I don't know how to explain these.
Best Answer
In Latin, "gn" was pronounced as two consonants
In Latin, the spelling <gn> represented a sequence of consonants or a heterosyllabic consonant cluster. While we don't know the exact pronunciation for sure, current scholarship leans in favor of the value [ŋn], a velar nasal followed by a coronal nasal. The history of the similar pronunciation [gn] (a voiced velar plosive followed by a coronal nasal) is a bit unclear, but it certainly was used in later time periods when Latin was no longer spoken as a native language. Even if it was pronounced as [ŋn], many modern phonemic analyses of Latin treat <gn> as /gn/, with an allophonic realization of /g/ as [ŋ] in this environment. Interestingly, the situation with <gm> doesn't seem to have been exactly parallel, as there isn't really any evidence for a pronunciation *[ŋm] existing in Latin. It may be relevant that Classical Latin "gm" mostly arose from relatively late processes of vowel syncope (e.g. tegmen < tegimen): ancient "gm" clusters seem to have changed in Latin to [mm], with assimilation in both manner and place of the first consonant to the second (e.g. in the word flamma "flame").
I wrote a more detailed post about the pronunciation of "gn" in Latin on the Latin SE site; my main source was W. Sidney Allen's Vox Latina.
In many Romance languages, Latin "gn" turned into a palatal nasal (a single consonant)
As far as I know, none of the Romance languages retains either [ŋn] or [gn] in inherited/popular vocabulary. (The sequence /gn/ does show up in a number of languages in some learned vocabulary from Latin or Greek; e.g. French ignition.) In Romanian, Latin "gn" became "mn" [mn] (a similar sound change affected Latin "ct", changing it to Romanian "pt" [pt]). In Western and Italian Romance languages, Latin "gn" has generally become a palatal nasal [ɲ] (in standard Italian, the palatal nasal is an "intrinsic geminate", meaning it is always pronounced as a long consonant [ɲː] when it occurs between vowels).
Some of the sources that I've read, such as Allen, say that [ɲ] points towards earlier [gn], via sound changes like [gn] > [ɣn] > [jn] > [ɲ(ː)], but honestly I'm not sure why we couldn't have gotten [ɲ] from [ŋn] via some process like [ŋn] > [ɲn] > [ɲ(ː)]. Anyway, I don't think the details of the sound changes leading up to [ɲ] are particularly relevant to your question. The important point is that Western and Italian Romance languages developed an alternative pronunciation for "gn" that was not the same as the pronunciation that had been used in Latin. Despite the change in pronunciation from Latin, the spelling <gn> was used for this new sound in various Romance languages (although as tchrist's comment mentions, not all: <nh> was also used in some languages, as it still is in modern Portuguese; modern Spanish uses <ñ>; and at one point in French the trigraph <ign> was used to represent [ɲ], a convention that has left a remnant in the modern French spelling <oignon>—see the French SE posts L'orthographe « ognon » a-t-elle une date de péremption ? and Oignon/ognon : découpage syllabique ? for more details).
Palatal nasals from sources other than Latin "gn" are nevertheless "unetymologically" (but regularly) written with "gn" in certain Romance languages (e.g. French, Italian)
The [ɲ] sound in Romance languages has other sources than Latin "gn". In your question, you've mentioned some words that exemplify this: French campagne and Italian campagna come from a Latin word ending in "nia". In words like this, [ɲ] developed from coalescence of /n/ with a following palatal glide, but the digraph <gn> is nevertheless used in Italian and French. This could be considered "unetymological", but it's not irregular: <gn> has become the regular way of representing the palatal nasal, regardless of etymology, in these two languages. Furthermore, Italian uses <gl> and <gli> to represent the palatal lateral, which is often not derived from Latin "gl": e.g. Italian foglio [ˈfɔʎːo] "sheet" comes from Latin folium.
The same development explains the <gn> in French ligne "line" from Latin linea (the "e" in the Latin form turned into a palatal glide, which then coalesced with the preceding nasal to form [ɲ]). The English word align is related to French ligne.
Another historical French sound change got rid of [ɲ] when it came before another consonant, replacing it with [n] (along with a palatal offglide on the preceding vowel). This is why the infinitive form feindre has "n" instead of "gn". The "d" here arose by epenthesis of -nr- to -ndr-. But the present participle stem feign- was not affected by either of these sound changes, because the sound [ɲ] here came before a vowel. The development of Latin /ng/ to French [ɲ] in certain contexts is a regular sound change, and the corresponding spelling change to <gn> is also expected, not irregular. I wrote an answer to a related question on Linguistics SE: How did OF. peindre derive from L. pingere, with a “-ng-” > “-nd-” change?
"arraign": unlike the other words mentioned, it doesn't seem to have come from a Romance word with a palatal nasal
The word arraign is the only example you gave where it seems impossible to explain the use of <gn> in terms of influence from the spelling of the French etymon. The OED says it comes from "Anglo-Norman arainer, areiner, arener, Old French arais-, areis-, aresnier < Latin adratiōnāre". Although some French words now spelled with <ni> seem to have had old variants with <gn> (e.g. the OED says <magnioc> existed as a 17th-century French form of manioc), I haven't found any evidence that this is the case for araisnier. So the use of <gn> in the spelling of this word does seem to have originated in English, by analogy with the spelling of other words.
As your question mentions, two other words that seem to be spelled with <gn> in Modern English for similar reasons are sovereign and foreign.
I don't think the use of "y" in Middle English spelling is very relevant to the use of "gn" in Modern English spelling
In Middle English, Y was used in many of the same ways as I. Apparently, Y was particularly likely to replace I when the vowel was "long", or when the letter was next to one of the "minim" letters M or N (sequences like im, in, mi, ni are supposed to have been avoided in English writing at certain time periods because of the similarity of the strokes used to write these letters). (Complete Works Of Geoffrey Chaucer, W.W. Skeat) So I think feynen and feinen or alynen and alinen are basically equivalent spellings.