Prudent or well-advised.
The sense is now archaic. Some set phrases like "it is well to" meaning "it is a good idea to" or "it would be well to", etc. still have some currency.
Remember, Benedick is talking about how love makes people (especially men) foolish, and claiming that this won't happen to him; so while love makes some foolish he remains prudent/well-advised because he does not fall in love.
what's the meaning of "I'll none"
"I'll have nothing to do with [whatever is the current topic]" "I'll have no part in [whatever is the current topic]".
This is perhaps the trickiest bit of this whole question; the rest can be found by looking at some of the more extensive dictionaries (The OED is excellent and wiktionary is good in some ways, certainly good enough to answer the rest of this question, and free) but none as a verb isn't listed because "I'll none" or "I'll none of it" was a set phrase of the time with this "I'll have nothing of this" meaning.
I'll never cheapen her
A literal translation into contemporary English would be "I'll not bid on her", "I'll not offer to buy her", "I'll not ask what her price is". Figuratively it means "I'll not try to woo her or seek her hand".
(Some later readers imagine this is cheapen as in "make worth less" applied to ideas of the value of chastity in a woman—which is after all made more of later in this play—and so see an irony in this coming in response to his insistence that any woman he might fall in love with be virtuous. However, this sense isn't attested until a few decades after Shakespeare died, so it's unlikely it was meant. Today though Shakespeare's sense of "offer to buy" is obsolete while the sense "make less valuable" is common).
The expression is may very well.
(The be is part of the progressive infinitive verb be learning.)
May here refers to possibility. The young man or the coed may be learning..., but they may not be. It has a higher possibility than might.
Using very well with may (be learning) can mean several things. It really depends on the speaker's attitude toward the action and/or the tone of voice. It could mean the action has a lesser, equal or greater possibility of happening than indicated by may alone.
"It may rain" = "It's possible that it will rain".
"It may very well rain" = "It's quite [possible] that it will rain".
(From Word Reference, link in same thread as below).
It could be a concession by the speaker:
It may very well rain today, but I'm going to play tennis anyway.
I concede that there's a good chance that it may rain today, but...
Since the next sentence of your text starts with a but clause, it has this similar concessive meaning.
Lastly, in general terms, it could be used ironically, indicating that the speaker doesn't think it's going to happen:
may very well" is sometimes used cynically, followed by "but", as in "What you say may very well be true, but I'm not convinced". In this case, "very well" is being used somewhat ironically, as the speaker really thinks that what you say is not very likely to be true.
(Word Reference).
Best Answer
The OED attributes this now rare construct to Scottish and Irish English (northern) with the sense of 'expressing surprise, anticipation, resignation, or acquiescence'. It gives among several examples:
The old woman tried to comfort her, beginning with her accustomed, ‘Well-a-well!’ (Elizabeth Gaskell, Mary Barton, 1848) and Tam simply wad say, ‘Weel-a-weel, I'll jist by your counsel be guided.’ (David Willox, Poems, 1898).