Cisco ASA: How to route PPPoE-assigned subnet?

Solution 1:

How does the ASA know which range it is responsible for on the Fiber interface?

It doesn't. Your ISP knows (hopefully) that hosts within given sub-net are reachable via the ASA. In case the ASA wouldn't have connected peer (or other proper route to), the ping-pong would begin: ISP sends packet to ASA, and the ASA sends it back by its default route. That's why I'd recommend at least adding static route saying that your assigned subnet is reachable via Null device.

How do I use the addresses from my range?

Plenty of ways actually. For e. g., you can NAT your private IPs to some IP within the subnet. You can subnet it further and connect some hosts to your ASA directly on those IPs.