OK, so I can only speak to 14th-15th century England, but there were a LOT of channels through which people could potentially learn about the Bible, and a really determined illiterate person, like Margery Kempe, could acquire some very sophisticated knowledge, enough to cite specific – and not especially famous – passages to defend herself against heresy charges.Some scholars think Kempe was lying about or exaggerating her inability to read in English in order to look less like she was interpreting the Bible for herself, but she definitely couldn’t read Latin, and she mentions getting a priest friend to read books to her, including a Bible with commentary. This might be a Latin Bible that he translated for her, or it might be a (banned) Middle English Wyclifite Bible, we don’t know for sure. Importantly, the crackdown on English translations only occurred after they had been circulating for a few years, and it wasn’t consistently enforced, especially if they were in the hands of the “right” people. (This is why we have surviving copies of the Wyclifite Bible – they were often hiding in plain sight. Many of the ones we do have are incomplete – e.g., just the New Testament, or the Gospels alone, or Gospels plus selected parts of the Old Testament, with Psalms and Proverbs being particularly popular. This probably tells us a lot about which parts of the Bible people thought were important!)
There were also authorized, church-sanctioned retellings of Bible stories in English, some of which would have been accessible to the illiterate, like mystery plays. For those who could read, there was also Nicholas Love’s Mirror of the Blessed Life of Jesus Christ, which was a retelling of the Gospels intended for lay people. Love spends lots of time conveying the Church’s “official” interpretation of particular passages, and he smooths over the contradictions between the different Gospels and knits them together into a single story, but you would still learn a lot about what the Bible actually says if you had access to the Mirror.
And, most importantly, there were English-language sermons, which weren’t always part of the regular church service, but popular preachers drew huge crowds. Sermons often involved translating and explicating particular passages, as well as storytelling in general (not all of the stories would have been from the Bible, but many of them were). It’s fictional, but the sermon in lines 483-659 of Chaucer’s Pardoner’s Tale is a plausible example of how a traveling preacher would have used the Bible: he cites lots of biblical examples – Lot, Herod, Adam, etc., clearly assuming his audience will recognize the stories and names, as well as a few direct quotes from St. Paul and from the Ten Commandments; at one point he pauses to specify that the Bible character he’s talking about is the obscure Lemuel and not the more familiar Samuel; but many of these examples are out of context or used in service of a slightly idiosyncratic point, such as treating the Adam and Eve story as a warning against gluttony. The Pardoner isn’t meant to be a good example of a preacher – he’s all kinds of crooked and self-serving, and most clergy were probably more scrupulous about HOW they used their biblical source material – but their general approach, and the level of prior knowledge they expected from their audiences, were probably similar.
ETA: So, as to the specific questions in the OP, I’d say the most likely answer is “yes to the more salacious bits” (the Pardoner expects his audience to know that Lot slept with his own daughters), but “no to obscure prophets,” unless, like Jonah, they come with interesting and colorful stories.