One meaning of the word "draw" is to pull in various directions, as in a horse-drawn carriage, i.e, one pulled along by a horse. Or consider that a chimney sweep will tell you that
Chimneys often draw at least a small amount of air, even when
there is no fire below....
That is, a chimney will pull up air from the fireplace room and out of the house even when there's no fire lit.
In a figurative sense, if the meaning is pull towards, then the person feeling the force of the pull will feel an attraction. Thus from Terry Brooks' The Elf Queen of Shannara:
She could feel the draw of two choices, very different, each
compelling.
Two choices, each with its own attraction. And that's the meaning of the sitcom line, "What's the draw?" namely, "What's the attraction of having children around?"
I think there is a clue to this just a little farther in the passage itself, where the speaker goes on to mention his "state"... and I think it is in the notion of "statehood" that these expressions (and perhaps their understanding) are to be grounded.
That is, the speaker is speaking from the state of not just being a man, but man, as in ALL men, and further, ALL (of) "man", that is, a state of mind not just identifying with all the members of that given set, but to represent them singularly, collectively, and superlatively (cf. expressions like "I am woman, hear me roar!").
Additionally, there is an allusion to "man" being used here as an abstraction, not just a concrete representative of same, or all of them, but the whole idea of being a man (or woman, later), as in expressions like "I am VENGEANCE," or similar identifications with abstractions of concrete things, or abstract concepts themselves. This identification-with-the-concept is what gives the speaker the presumed mandate to speak on that entire concept's behalf, as he indeed goes on to do throughout the quoted passage.
However, as the somewhat more modern examples above suggest, I am not sure this is so much a feature of Shakespearean English, but rather, that Shakespeare's prose simply uses more of this forceful stuff. Indeed, I would bet you a shirt frill that if H.G. Wells dumped you in ye local publick house in Merry Ole Stratford-Upon-Avon, you'd be hard pressed to hear any of the locals express such prose while quaffing their brews.
But that, of course, is pure speculation on my part. :o)
Best Answer
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.
"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.
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).