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"There has been a fair bit of work done on AC-AC adapters, and a lot of graphs that show their output waves. But if they are switching mode power supplies, how do we know their output has anything to do with the input they receive?"

No, they are not. If you had read the report in Building Blocks, you would know exactly what they are and their properties.

Although "doorbell" transformers have been used by some, I believe it is likely that they might be on the edge of saturation (in order to minimise manufacturing costs) and therefore might suffer unacceptable phase and amplitude errors that are voltage-dependent and thus variable to an unacceptable degree. If you can satisfy yourself that the output is a sufficiently accurate representation of the supply waveform for your purposes, then there's no reason not to use that one. You then of course need to choose the output voltage and adjust the divider ratio if necessary to give approx 1.1 V (for the emonTx) or 1.66 V (for a 5 V Arduino) at the analogue input when the supply voltage is at its maximum.

If you are using it with the emonTx V3, then further considerations apply and unless it's relevant properties are very close to or better than those of the 'shop' Ideal adapter, then proper operation of the emonTx V3 is not guaranteed.

There is insufficient information in the link you provide for me to have any idea as to whether that model would be suitable.

The transformer exhibits a phase lead that varies from 4° at the lower limit of the supply range up to 7½° at the upper limit. 

This doesn't sound like a standard transformer to me. In any case I need a 120v/60Hz solution, and wall wart style AC-AC adapters are not anything I can easily get my hands on.

Saturation may be interesting and something to look out for, I'll have to read up on it. 

"wall wart style AC-AC adapters are not anything I can easily get my hands on"

On the contrary: http://shop.openenergymonitor.com/components/

"This doesn't sound like a standard transformer to me."

Saturation isn't the only aspect of transformers that you need to read up on, then.

I really like the idea of having a single transformer aka "wall wart" but the power limitations of the emonTX V3 circuit don't work for my power draw (50mA min).

Has anyone experimented with a Split transformer, like http://www.digikey.com/product-detail/en/F10-110-C2-B/237-1557-ND/3986429 and feeding direct AC onto your board, or to a sub-board?  

I am thinking I could use one leg for AC monitoring, rectify the other leg and use a regulator to feed my circuits. 

Would the draw on one leg pull down the voltage on the other leg enough to throw off AC measurements? 

Thanks in advance.

Whether the load on one secondary winding is reflected into the other depends on the properties of the transformer that are rarely published, not even a basic property such as the regulation (which would give us a good guide) is mentioned on that page. So unfortunately the only way will be to build it and test it.

[Edit: The data sheet gives the regulation as 25%. That's not promising.]

Understanding that 0% is perfect, yet impossible, and the Ideal adapter is 27%.  What would you consider to be an acceptable level?  I am blasting through datasheets and when listed the best I have found is 20%.

You won't find a small transformer that is much better than that. Power distribution transformers will be down in 2-5% range. (Imagine your supply voltage shooting up by 5% per transformer all the way back to the power station when the system load decreased by 20%.)

You'll be able to use a full-wave rectifier of course, and it will help if you have lots of excess voltage so that you can afford to limit the peak current into the reservoir capacitor and thus spread it out over a longer time, as we did in the V3 power supply.