Electrical – Trying to find the transformer for the doorbell

OK, this may be a side-effect of the way the Ring powers itself.

The Ring is in series with the chime. In series circuits, the same amount of current (amps) flows around the loop through both devices. And by the way, this problem crops up with Nest thermostats and dimmers too.

How does the Ring power itself? By deliberately letting a little bit of current flow through the circuit, including through the chime. The Ring (like the Nest and dimmers) is betting that a small enough current won't make the chime (furnace, lamp) activate. And it is guessing at how much current it can get away with.

So my guess is: you have a chime which is more efficient than most. Sure, when the visitor rings the bell, the Ring shorts the two wires to simulate a doorbell push, and then it lets go of that correctly. And then, it needs to recharge its internal battery - transmitting video over WiFi is hard work. So it allows current to flow at that lower rate. Unbeknownst to the Ring, that is enough to ring the chime. The Ring isn't timing out when it stops; it's just finished charging.

Now like I say, smart thermostats have the same problem: they charge by sending some current down the W line, hopefully not enough to engage the furnace relay. If that doesn't work, they give you an option to hook up the "C" line to power the thermostat directly. Unfortunately it does not appear the Ring has an option like that.

TLDR: Your chime is too efficient for the Ring to vampire-charge itself. Get rid of one.

Well, if you're in love with that chime, there might be another option, but it's definitely not UL-approved. You could alter your chime to be less efficient... by putting a resistor of appropriate size in parallel with the chime. Finding the appropriate size would take some measuring or experimentation. You would not want one which gets too hot, worst case someone holds down your doorbell button for a long time. A fellow shouldn't be able to burn your house down by holding the doorbell button.

Yes your selecting an AC voltage setting is correct for a typical doorbell transformer.

If you can find the transformer itself the first test would be to remove one or both low voltage wires from the transformer output and then measure the output with the meter. This will tell you if the transformer is OK or if there is some problem with the AC line voltage feed to the transformer or the transformer itself has failed. This isolation could save a good amount of time trouble shooting the low voltage wiring at the doorbell and switch.