To begin with,
Dear Mrs. Smith,
at the start of a letter, is a special case. This is a formal way of opening a letter, and does not imply any romance or sentiment. It would not seem unusual or ironic to most English speakers to read a letter that said:
Dear IRS, I think you are a cancer and a blight on humanity. Love, Johnny Greenink.
That is just how many people open and close every letter, whether the recipient is really "dear" or whether they really "love" them or not.
Used as a term of personal, as opposed to postal, address, the differences are going to be primarily based on regional dialect. Whether you hear:
Dear, could you pass me the sugar?
Darling, could you pass me the sugar?
is going to depend on where you're listening. In the American South, you might hear "darling" used for a spouse, a child, or even a stranger. In the Midwest, you would be much more likely to hear "dear." In California, you are more likely to hear "honey" than either, in my experience. It's more idiomatic to use the name or the nickname, not both.
However, in most dialects, using either "dear" or "darling" for strangers has an unusual association; it is considered either archaic, affected, or lower-class, not to mention sexist when used by a man to a woman. Some other form of address for strangers is always preferable.
When used as an adjective, as in
This is my dear wife,
there is more of a marked difference, at least in the mid-Atlantic American English I speak. I would be very unlikely to use the phrase "My dear wife" unless I were addressing her directly in a very formal situation:
To my dear wife, Edna, I bequeath my collection of Dresden china figurines.
You might see someone write something, or deliver a line in a political speech, like:
My dear husband is good at math, but terrible at counting calories
but it would sound very odd in informal speech.
"Darling," on the other hand, carries a much more informal, "oversharing" kind of implication. If you heard someone say:
I met him, and I met his darling wife, and his three darling children
you are immediately going to jump to some conclusions about that person. What conclusions will depend on dialect, tone, or context, but at a minimum it is going to be highly informal. In the UK I believe this would be a middle-class or possibly upper-class marker; in the U.S. it would sound pretentious or lower-class; in the U.S. it is more likely to be spoken (in that context) by a woman or a gay man. What I'm saying is, "darling" is a word with a lot of baggage, so be careful with it.
In my dialect, the most idiomatic way to say your last two examples would be to leave out all of the "dears" and "darlings." You could use "lovely" in your last example:
May I introduce my lovely wife, Edna?
but even that would be (1) a cliche, and (2) condescending. Just call her your wife and be done with it.