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 Today though Shakespeare's sense of "offer to buy" is obsolete while the sense "make less valuable" is common).