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I can confirm that the message is accurate. I just switched to the 'Swipe' style lockscreen and was indeed then unable to use my fingerprint to log in to Dashlane or authorize purchases in the Play Store. Enrolling a fingerprint requires selecting a backup screen lock (pattern, PIN, or password). It doesn't seem like you can forego the lockscreen and still have fingerprint convenience.
It's not a bug, it's very intentional.
The short answer is that fingerprint data has to be stored on encrypted storage, and the OS uses your lockscreen authentication as a part of the key to access encrypted storage. (The more technical answer is that the fingerprint data can also be stored on an entirely separate bit of storage/processing capability, but the OS may still enforce the encryption requirement for good measure). You can read a bit more about how fingerprint data is handled here.
Is there a particular reason you don't wish to use a secure lockscreen and thus benefit from numerous security features built into your new phone? The speediness of Google's fingerprint auth makes finding a good balance between security and convenience a no-brainer.
No, because that is how it is designed. Fingerprint is a screen lock which can also be used by other things if it has been set up. It is not something that can be set separately from the lock screen, so that is not going to happen.
You can set a delay of up to 30 minutes between the screen timing out and locking (try the gear icon next to the screen lock method - a slightly inconsistent UI for this), so if you could find a way of making that longer that would give you something functionally similar to what you want. Whether that is possible, and whether it's possible without root, I can't say. I had a quick flick through Tasker and the Secure Settings plug in and didn't spot anything obvious.