Automatic tracking of monitors in Health.app? Unless I am mistaken, HealthKit/Health.app will automatically track some BlueTooth sensors, like heart rate straps? Is that true?

Is it also true that other sensors do not work this way, and only work when a 3rd party app is running. Is that also true?

If so, is there a list of such "auto track" devices?
There are no "auto-track" devices per se. You have to go in and give certain apps or devices access to the Health data. 

As detailed on Apple's website, you need to choose the fitness app or accessory you want to use, and this will connect it to your Health app.

When you connect apps as a source of data to your Health app, they should automatically push in the data, but you might have to open the app for it to trigger. On the other hand, there are many, many devices that support automatic synchronization, like the Garmin Forerunner 220. 

Unfortunately there are too many devices supporting synchronization with an iOS device, but most apps or devices that are capable of synching with the Health app support auto-synchronization when in close proximity to the device.

You can see a list of HealthKit compatible devices here: http://www.applehealthkit.com/compatible-devices-apps/.