You have pretty much already answered your own question, but here it goes for completeness:
1. Yes, your Kindle will try to update its firmware if it's registered and connected via Wifi.
2. To get a fully-functional Kindle which simply does not auto-update, register it and use the BackDoorLock hack.
3. Of course, if you always keep the Kindle in Airplane mode, it also won't update, because it can't download the updates. The obvious downside is that you also won't be able to use the Kindle store or browser.
4. If you fake-register the device (yes, the registration essentially is a text file), it'll work like a normal one. But once you go online, the registration info will be removed and the device will turn back into an unregistered one.
5. If you don't register it at all, you will lose some functionality (like manipulating collections using the official UI, and I believe also things like setting device passcode etc.)
6. Collections Manager works independently of the device's registration status.
So, points 2.-5. are more or less "in decreasing order of usability/usefulness".
My suggestion is to install the BackDoorLock, then register the device. You may still want to keep the device offline most of the time if you're concerned about battery life and/or privacy, but that's your decision.
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