I did some research using the Corpus of Historical American English to see if I could track the history of these words. Each of these words has a different story to tell.
DREAMED and DREAMT
See the raw data (Google Docs)
In the early 1800s, dreamt was more common than dreamed but by the mid-1800s, dreamed was much more common and has stayed so since. While there is nothing wrong with continuing to use dreamt, dreamed is definitely the more common form.
LEAPED and LEAPT
See the raw data (Google Docs)
Leaped has long been more popular than leapt, though leaped has been in decline since 1900, and leapt has been on the increase since 1950, and today they are about equally common. It is likely that if the current trends continue, leapt will become decisively more common than leaped within a decade or two. Indeed, in the Corpus of Contemporary American English, leapt has 484 incidences for 2005-2010 and only 460 for leaped. So, both are about equally common these days and you are in good company if you prefer leapt.
SWEEPED and SWEPT
Neither COCA nor COHA have any results for sweeped, nor does any dictionary I checked list sweeped as a possible past tense form for sweep. Sweeped doesn’t appear to have had any currency in American English since 1810. Google reports only 43,000 results for sweeped compared with 78,000,000 for swept.
LIGHTED and LIT
See the raw data (Google Docs)
Apparently, lighted was much more popular than lit, from the early 1800s until about 1940. During this time lit was steadily gaining popularity, while lighted began a precipitous decline in 1940. Today, lit is much more popular than lighted, so if you prefer lit, you are in very good company. However, there would be trouble objecting to lighted on historical grounds, as lighted was by far the most common form until the 1940s.
It is a poetic way of replacing the word work in a gender-specific way. To wit:
When the quotation says that Adam delved, it is saying that he delved into the soil of the earth he farmed; this illuminates the connection with the primary meaning of delve. In other words, it is another way to say that he tilled the soil, an old male-specific metaphor for general work in English.
When it says that Eve span, it is similarly referring to a female-specific metaphor for work; not so long ago, women were largely responsible for tailoring and mending the clothing of their household, and even today hobbies like crocheting and knitting are strongly associated with the feminine sex. Weaving cloth in those days would have required a distaff and a spindle; a spindle is a weighted object that spins as it releases cloth to the loom, which explains the use of the participle span.
Though you didn't ask about it in your question, gentlemen had less of a vague meaning than today, where it can be applied to just about anybody, rich or poor; back then, in John Ball's time, to have been called a gentleman would have been to be a man of means, of good family and distinction, and it would connote more of a class divide than we observe with the word today. Indeed, gentlemen has a shared etymological root with the word gentry, which as you'll note still retains that distinction for scholarly use.
So in other words, the quotation can be crudely paraphrased thus:
Back when our righteous forefathers worked, who among them would have been better than the other, who would have observed any class distinctions?
That massage would have soundly resonated with the rebels of the Peasant Revolt John Ball was preaching to.
17 And unto Adam he said, Because thou hast hearkened unto the voice of thy wife, and hast eaten of the tree, of which I commanded thee, saying, Thou shalt not eat of it: cursed is the ground for thy sake; in sorrow shalt thou eat of it all the days of thy life;
18 Thorns also and thistles shall it bring forth to thee; and thou shalt eat the herb of the field;
19 In the sweat of thy face shalt thou eat bread, till thou return unto the ground; for out of it wast thou taken: for dust thou art, and unto dust shalt thou return.
20 And Adam called his wife's name Eve; because she was the mother of all living.
21 Unto Adam also and to his wife did the LORD God make coats of skins, and clothed them.
Best Answer
Span is an older strong past tense form of spin — this is the past tense form that existed in the older Germanic ancestors of English. In German, for example, the past tense of spinnen is still spann. In English, span has mostly fallen out of use in favor of spun for both the past participle and simple past forms. This is a form of paradigm leveling. It has occurred in other words like sting (no stang) but not in ring (where we do have rang).
The OED actually gives your quote as follows:
Note that this quote is dated from 1560. At this time, span was a more common past tense form, and of course well-known quotes are usually more resistant to being updated to modern vocabulary and grammar. (Interestingly, it looks like dalve has been out of use long enough by now that, at least where you read it, it was updated to delved.)
The OED cites uses of span dating up to the late 19th century, for example, this quote from 1882:
So, your answer is: if this sentence were constructed today, then spun would be the correct word, as you suspect. This is simply an old quotation where that word remains fossilized.