Let me answer the last question first. Yes — oxes is liberally attested in the Middle English period, with its veritable explosion of different word forms and spellings, and oxes has cropped up occasionally ever since. The form foxen has occasionally been seen as a plural of fox, as in David Booth’s Analytical Dictionary of the English Language . But as a plural, it would have had stiff competition from foxen = vixen “a female fox”. When it has appeared as a plural, I would guess it was under the pressure to match oxen. Or it may show a Dutch influence. The examples Booth gives — housen, foxen, eyen — have the Dutch forms, huizen, vossen, ogen.
Let’s take a closer look at the provenance of the two words. First, the obligatory litany of cognate forms. Now come on, no complaining; you knew this was coming. :)
The word ox comes from Middle English ox, from Old English oxa. Cognate are Old Norse uxi, Old Saxon ohso, Middle High German ohse, Old High German ohso, Welsh ych (pl. ychen), and Gothic aúhsa — all (well, not the Welsh) from Primitive Germanic *uhsōn.
The word fox comes from ME fox (and southern dialectal vox), from OE fox. Cognate forms include ON fóa “fox” (“vixen”, actually; the noun is feminine) — ON borrowed the OE fox, unchanged, but it was only used (and only rarely) in the metaphorical sense of a kind of fraud; Norn fūa; OS fuhs, vuhs; MHG voha, and vuhs; OHG foha, and fuhs; and Gothic faúhó — all from Primitive Germanic *fuhsaz.
Great, you’re saying, but how does this help us understand why the plural forms differ? Well, take a closer look at the OE forms, and you will notice that one of them ends in a vowel. Our ox originally ended with a vowel in all the Germanic languages, as it still does in Modern Swedish oxe, German Ochse, and Frisian okse. But in Modern English (as in Modern Dutch os) the terminal vowel has been lost. But fox had its terminal consonant from the beginning. OE fox is a strong noun, hence its plural form is foxas; but OE oxa is a weak noun, hence its plural would be … anyone? … oxan. And there you have it: that’s the reason for Modern English oxen, but foxes.
Are there other words like these? Not pox (actually = pocks, already plural). Nor tax (from Latin via Old French).
How about box? Like ox and fox, this is a direct survivor from OE box, but this word is a feminine strong noun, so its original plural, buxa, was of a third kind! Subsequently, buxa was standardized by force into boxes, just as oxen will probably someday become oxes.
And what about lox? This comes to Modern English via Yiddish, which is not Hebrew, but basically German. There is actually an all but forgotten OE antecedent in lex, leax, læx “salmon”. This was a strong noun (like fox), so its the plural would have been leaxas. But strangely enough, modern lox has no plural!
Any others? How about two more, just for fun. You are having fun, aren’t you? Modern English ax(e) is from OE æx, æcs(e); its correct plural is like box, æxa, but similarly, it has been “standardized” into the form, axes. And wax is from OE weax, wæx; a strong noun like fox, so its plural is weaxas. We’re all right there with Modern English waxes.
What does any of this have to do with Tolkien (as everything must :)? I’ll leave you with two quotations (emphasis added):
A fox passing through the wood on business of his own stopped several minutes and sniffed. ‘Hobbits!’ he thought. ‘Well, what next? I have heard of strange doings in this land, but I have seldom heard of a hobbit sleeping out of doors under a tree. Three of them! There’s something mighty queer behind this.’ He was quite right, but he never found out any more about it. — The Fellowship of the Ring (Chapter 3, “Three is Company”)And:
In Dasent’s words I would say: ‘We must be satisfied with the soup that is set before us, and not desire to see the bones of the ox out of which it has been boiled.’ [...] By ‘the soup’ I mean the story as it is served up by its author or teller, and by ‘the bones’ its sources or material—even when (by rare luck) those can be with certainty discovered. — “On Fairy-Stories”
 Booth, David. An Analytical Dictionary of the English Language. Volume 1. London: J. and C. Adlard, 1830, p. xxxiv.