Punctuating question tags: A question mark is always required, isn't it. (Well, isn't it?)

I'm not sure it counts as a "definitive answer", but so far as I'm concerned punctuation can't be used to differentiate OP's two intonations (and hence, meanings).

The standard rules of punctuation require the question mark to follow all constructions framed as questions, though style guides generally make exceptions for either/both of these types:

The teacher was waiting for them when they got back and was she mad! (really an exclamation).
Miss Kate, will you take a seat over there in your chair again. (courtesy question/invitation/order).

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It may be worth noting that by introducing an italic font as well as punctuation, some other interpretations can be implied...

"You didn't leave the dog in the car. Did you?"

...which with no other context, implies the speaker is addressing two people - stating/confirming that the first didn't leave the dog in the car, and asking the second whether they did.

"You didn't leave the dog in the car, did you?"

...which implies OP's "former" case (speaker is concerned/incredulous), but shifts the element of surprise to the fact that the dog might have been left in the car, rather than somewhere else. One could just as easily italicise/emphasise the word "dog" (surprise that we're talking about the dog, rather than, say, the baby).

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Those contrived contexts are probably only two of many alternative meanings that can be conveyed by how the text is written, but the level of diferentiation OP seeks simply doesn't exist.

I don't normally answer my own questions, but in this case I feel compelled to do so. FumbleFingers' answer — the only answer this question has received* — while well-argued and not incorrect, feels like the answer of someone who is faced with a problem he recognizes but cannot solve, and so falls back on whatever has served in the past: in this case, the "standard rules of punctuation."

I've thought about this for a long time, and I am coming to the opinion that "standard rules of punctuation" are a fluid concept, changing with the rest of language. Certainly no one would suggest that the punctuation rules of the 18th or 19th centuries would apply today. The meaning of entire statements in the U.S. Constitution, for example, would seem to turn on archaic styles of usage for commas. Just look at the Second Amendment:

A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed.

By today's standards, there are too many commas. The extra commas have created a storm of controversy. We would not write the same sentence the same way today. Punctuation has moved on.

I am going to suggest, then, that punctuation may be used (to some, abused) as a matter of expression. Writers have been doing this for many years. Normally we would suggest that dialogue in text, say, should be governed by "standard rules of punctuation"; the statements made by characters are surrounded by quote marks, and there are rules governing what is done with other punctuation inside and outside the quotes. But then we look at works by James Joyce or William Gaddis, and we see that some authors, at least, decided that an em dash was a better way to set off instances of dialogue. Here is a fragment from Joyce's novel A Portrait of the Artist as a Young Man:

Stephen had answered: Stephen Dedalus.
Then Nasty Roche had said:
—What kind of a name is that?
And when Stephen had not been able to answer Nasty
Roche had asked:
—What is your father?
Stephen had answered:
—A gentleman.
Then Nasty Roche had asked:
—Is he a magistrate?

It is worth noting that Joyce doesn't even use an em dash in the first line. He uses a colon instead. Similarly, Faulkner tried to move English forward by eliminating apostrophes from some of the contractions (notably, dont, wont, cant and aint). Now, you may argue that none of these treatments caught on, but that does not negate the work of these authors, nor call into question their right to make their own rules in the pursuit of expression.

At this point I have to remind everyone about one of the conditions I cited in my question above:

But in written form, especially dialogue, it feels to me that question tags meant to be spoken with a falling intonation might get by with just a period

So — again, reminding everyone that we are talking about writing as an expressive medium, and that nobody has addressed this aspect of the question — I am going to conclude that using a period instead of a question mark at the end of a question tag may be construed as a matter of stylistic expression. I wouldn't do that in an academic paper, or a scientific treatise, of course, but we are much less likely to encounter dialogue in those venues anyway.

^{* As of the original posting date; obviously that's changed now.}

I basically agree with FumbleFingers, I say you should always use a question mark, but you can't derive the intonation from the text so written.

But why would you restrict yourself when using the written word? Something like:

"You didn't leave the dog in the car, did you?", she asked incredulously...

or a slight rephrasal:

"You left the dog in the car, didn't you?", she asked incredulously...

would provide further context, which would then inform the reader regarding intonation (were they to read the written text aloud for instance).

Short answer - the written word requires more context, what stops you from providing it?

While it's ultimately a matter of style and preference, I think the case for the period is very strong. The period makes the writing clearer without becoming a distraction, and that's what good punctuation should do.

Using italics to distinguish the two kinds of tag questions works, but adding more punctuation creates more distraction.

The only real argument against the period that I'm aware of is that the punctuation "error" becomes a distraction. But in my experience this is not the case -- readers get used to it very quickly.

I think there's no way for punctuation alone to represent with absolute fidelity what the speaker's tone, intention, and exact meaning are in such a case. That would be expecting far too much of a simple graphical dot or squiggle.

The writer must decide just how important it is to convey one and only one particular precise tone and intention. The more important it is to convey the exact intended meaning, the more important it becomes to use contextual clues to get that meaning across to the reader.

Creative writing gives you license to describe the setting, the demeanor of the participants in the scene, the exact circumstances surrounding the statement at the moment it is spoken, and much more, in order to give a clear sense of how the statement and its tag question are being delivered. You could even explicitly describe the tone of the person's voice if necessary.

So, no, you are not required to use a question mark, but you are free to utilize all a writer's tools to make the sentence work the way you want it to.

"You didn't leave the dog in the car, did you."

This is a more concise way of expressing the following:

"You were supposed to leave the dog in the car. I don't yet know for sure whether you did or did not leave the dog in the car, but I have good reason to believe that you didn't. If this turns out to be the case, I will be very disappointed/upset, and dire consequences will probably result."

The intonation of "You didn't leave the dog in the car, did you." is similar to that of a declarative assertion. The emphasis will be greatest on "did", and there will also possibly be emphasis placed on "dog" and "car".