Use doorbell output (12VDC) to power arduino?
This seemingly simple task is actually quite complicated; a good challenge that's not impossible or easy to overcome.

The AC/DC transformer in your link only provides "12VDC/6mA"; not enough to run the Arduino, chime, and button LED; so it likely needs upgraded to something beefier; you can't have too many amps on such a transformer.

Then there's your chime, which could use a lot of current (buzzer/bell) or a little bit (speaker-based chime tones). If it uses a lot, you should replace it to simplify the other connections. Remember that power goes through the chime to get to the button; otherwise the circuit would be incomplete and the ringer's LED would be dark.

The chime is activated by the resistance of the button lowering itself to the point where the chime gets enough power, no longer being current-limited by the diode, 2.2k resistor, and small LED. Very simple design, but not ideal for easily intercepting.

So what's the easiest way to accomplish your task?

Looking at it closely, it seems the basic circuit is there for you already; you have a power-consuming node on the network in the form of the button's LED. 

You need to replace/augment the LED on the ringer with the Arduino. The 2.2k R19 could/should be replaced with the Arduino as well; just make sure your LED isn't over-drawn when the Arduino is chugging at full-steam, and add additional inline resistance as needed to keep it safe.

The other problem is powering the MCU while ringing. You'll notice the LED goes out while ringing. While that's nice feedback to the user, it's not because the circuit designers were UX-focused, it's because the chime's simple circuit  pulls all the current out from under the LED, just as it will an MCU in the same slot. 

You can get around this using a super capacitor. I would use an LM7805 in front of the Uno to deliver VIN +5 V instead of 12 V. Then you can "short" the Uno's GND to VIN with a 1 F 5.5 V capacitor ("F", NOT "uF"). Those caps are (amazingly) available for just a few dollars on eBay. 

So, at the end of the day, you basically just need to "upgrade the LED" and maybe upgrade the transformer. You could also re-create the switching pattern on your ringer with diodes if you're handy with circuit design, but if i were you, i would just "bodge" a few wires onto the existing PCB, and possibly alter that resistor. 

Good luck; sounds like a fun project and a good challenge.

EDIT:
I realized you might not need any circuit mods at all; a hall-effect current sensor anywhere the line, or just paralleling the chime terminals with a voltage-divider input to the Uno might suffice, if you didn't need the MCU physically at the button. Heck, a microphone sensor taped inside the chime could accomplish the task of detecting doorbell rings...