There are indeed many traditions that share elements of these myths, suggesting that they do have a common origin that is pre-Indo-European and has its oldest attested roots in Sumerian legends tied to some of the most ancient Sumerian kings. And, the element of not eating or drinking anything in the Underworld or Otherworld may be reinterpretations of the way in which the Sumerian god's rites were was ritually observed (with a taboo of not eating ground food during the ritual period in honor of the fact that a god's very bones were ground in the Sumerian myth).
Both of these legendary traditions are pre-Christian, and both were parts of the cultures of linguistically Indo-European people at some point. But, they seem somewhat remote from each other. While the story of Hades and Persephone is deeply rooted in a polytheistic pantheon, the notion of Faerie can be almost animistic.
Animistic religions are sometimes seen as a stage of religious development often seen as associated with a "tribe or a band society," before it reaches the kind of chiefdom society (and in particular, the somewhat federal late chiefdom phase "complex chiefdoms") which parallel the organization of dieties polytheistic ruling band in early Greco-Roman mythology and Norse mythology and were prevailling when these polytheistic schemes originated. Monotheism, in turn, can be associated with formative eras in some of the earlier centralized bureaucratic states - the Jewish state in the iron age Levant, the Egyptian state during the reign of King Tut, and the emergence of Christianity during a dominant Roman Empire, and the emergence of Islam as the tribal peoples of Arabia expanded by conquest to form a bureaucratized, urban empire.
The legendary culture of Faerie seems to live mostly in places that were once Celtic (Ireland, Britain, Normandy) and is sometimes described as Celtic folklore, although the word "Fairy" comes to English via Old French. It has its root in the Latin word for one of the Fates, but derivatives in other Romance languages apparently do not have the same connotations that they do in French and English. There also seems to be some sort of faerie tradition in Germanic Northern Europe. Wikipedia (linked above) notes that:
Folklorists have suggested that their actual origin lies in a conquered race living in hiding, or in religious beliefs that lost currency with the advent of Christianity. These explanations are not necessarily incompatible, and they may be traceable to multiple sources. Much of the folklore about fairies revolves around protection from their malice, by such means as cold iron (iron is like poison to fairies, and they will not go near it) . . .
[Citing  Silver, Carole B. (1999) Strange and Secret Peoples: Fairies and Victorian Consciousness. Oxford University Press. p. 47. and  Yeats, W. B. (1988) "Fairy and Folk Tales of the Irish Peasantry", in A Treasury of Irish Myth, Legend, and Folklore. Gramercy. p.1.]
The pivotal role of iron in the fairy myths suggest an origin of the myths in current form not earlier than the Bronze Age to Iron Age transition, which roughly coincides with the point in time at which Indo-European Celts and Germanic people emerge and expand in Western and Northern Europe, perhaps carred by these cultures, but perhaps as a legacy of a substrate pre-Celtic Bell Beaker culture. The notion of fairies as a conquered race living in hiding likewise fits with the notion of this lore a being the legacy of a conquered substrate people whose storytellers and holy priests who preserved these legends might have had to keep hidden and might have had to practice their Old European beliefs in private.
Basque mythology, which is the most solid continuous cultural link to the pre-Indo-European culture of Western Europe is arguably closer to the model of the Faerie legends than those of Classical Greco-Roman and Norse pantheons of Gods. For example, Basque mythology unlike Greco-Roman and Norse pantheons is a chtonic one "all its characters dwell on earth or below it, with the sky seen mostly as an empty corridor through which the divinities pass," in contrast to the Greco-Roman tradition situating its Gods on a sky oriented Mount Olympus and a comparable arrangement in Norse mythology. The mythological figure Mari in Basque mythology, likewise seems more of a faerie queen than a Hera to a male Zeus. This argues (weakly) for Faerie as a pre-Indo-European substrate absorbed into Celtic culture, and perhaps also into Germanic culture in a parallel cultural transmission.
On the other hand, folktale accounts of faerie in the English speaking world, at least, are generally situated in the era of conversion to Christianity in the early Middle Ages. However, to the extent that faerie represents an early iron age Indo-European Celtic and Germanic incorporation of a Western and Northern European substrate, perhaps with origins in Bell Beaker traditions, a placement of key tales at the point when Celtic and Germanic pagan cultures were superseded by Christianity is not necessarily inconsistent with this hypothesis.
Persephone's myth was incorporated into the polytheistic Indo-European pagan tradition and has been compared to similar myths "in the Orient, in the cults of male gods like Attis, Adonis and Osiris, and in Minoan Crete." But, this particular myth's origins seem to have roots that predate the arrival of Indo-Europeans in the Aegean.
The cult of Attis, the consort of Cybele is sourced in Phrygian mythology ca. 1250 BCE and later adopted from it into Greek mythology. Osiris has origins in Egytian legend. He "was at times considered the oldest son of the Earth god Geb, and the sky goddess Nut, as well as being brother and husband of Isis." Both of these cases parallel the Basque pairing of chtonic Mari and her consort Sugaar, and the story of Persephone and Hades could be seen as a gender reversed version of the tale for a patriarchal society displacing one in which women played a more dominant role. The origins of the Adonis myth are hotly disputed and there are contradictory accounts.
But, the existence of something similar to the Persephone myth in Minoan Crete also points to an origin of the core of this myth in a pre-Greek substrate rather than as a shared part of the Indo-European tradition. Its presence in the myth of Egyptian Osiris likewise suggests its place in a pre-Indo-European tradition that spanned the Mediterranean basin. Adonis and the other gods in this cluster of thematically similar dieties likewise shows strong similarities with the Sumerian God Tammuz:
In Babylonia . . . Tammuz, who originated as a Sumerian shepherd-god, Dumuzid or Dumuzi, the consort of Inanna and, in his Akkadian form, the parallel consort of Ishtar. The Levantine Adonis ("lord"), who was drawn into the Greek pantheon, was considered by Joseph Campbell among others to be another counterpart of Tammuz, son and consort. The Aramaic name "Tammuz" seems to have been derived from the Akkadian form Tammuzi, based on early Sumerian Damu-zid. The later standard Sumerian form, Dumu-zid, in turn became Dumuzi in Akkadian.
Beginning with the summer solstice came a time of mourning in the Ancient Near East, as in the Aegean: the Babylonians marked the decline in daylight hours and the onset of killing summer heat and drought with a six-day "funeral" for the god. Recent discoveries reconfirm him as an annual life-death-rebirth deity: tablets discovered in 1963 show that Dumuzi was in fact consigned to the Underworld himself, in order to secure Inanna's release, though the recovered final line reveals that he is to revive for six months of each year . . . Locations associated in antiquity with the site of his death include both Harran and Byblos, among others. A Sumerian tablet from Nippur (Ni 4486) reads:
She can make the lament for you, my Dumuzid, the lament for you, the lament, the lamentation, reach the desert — she can make it reach the house Arali; she can make it reach Bad-tibira; she can make it reach Dul-šuba; she can make it reach the shepherding country, the sheepfold of Dumuzid
The cult of Tammuz is referenced in the Hebrew Bible as part of pre-Jewish pagan practice at the door of the Temple in Jerusalem. Ezekiel 8:14-15. Dumuzi, in turn, can be associated with some of the earliest kings in Sumerian king lists, suggesting that legends around real historical figures in Sumeria may have spawned these myths."O Dumuzid of the fair-spoken mouth, of the ever kind eyes," she sobs tearfully, "O you of the fair-spoken mouth, of the ever kind eyes," she sobs tearfully. "Lad, husband, lord, sweet as the date, [...] O Dumuzid!" she sobs, she sobs tearfully.
A Sumerian source in the Semitic tradition is a good fit to the fact that much of the Books of Genesis and Exodus in the Hebrew Bible borrow from Sumerian myths of people who adopted Semitic languages. The centrality of Tammuz in Sumerian/early Semitic mythology may have facilitated the incorporation of this myth into neighboring non-Semitic forms of paganism, a religous structure well suited to borrowing myths from other peoples.
Thus, the myth of Persophone, may have made its way in the Pre-Jewish Semitic and ancient Egyptian traditions by way of Sumeria, whose copper age civilization gave rise to the oldest written documents and some of the earliest city-states before it underwent language shift to the Semitic Akkadian language. This same source myth could also have been a source for the Mari-Sugaar consort pair of Basque mythology, which in turn could have spread throughout much of Western Europe and Northern Europe as part of a Bell Beaker expansion if one accepts my own admittedly disputable identification of the Basque culture and language's ethnogenesis with the Bell Beaker culture.
The legendary element of not partaking of food or drink in the underworld does not appear to be part of the original Sumerian myth. But, it may related to the observed practice as late as the 10th century CE in Mesopotamia in which "Women bewailed the death of Tammuz at the hands of his master who was said to have 'ground his bones in a mill and scattered them to the wind.' Consequently, women would forgo the eating of ground foods during the festival time."
A prohibition of eating ground foods in Mesopotamia may have morphed into a prohibition on eating any foods in the underworld in the Mediterranean retellings of the story as it spread in a commom form (probably in a form close to the older Basque legends of Mari and Sugaar) that were absorbed independently from substrate cultures in Greece, in Germanic Europe, and by the Celts, in the latter two cases as part of a not quite polytheistic pantheon faerie tradition.