Not at all. Steve and Tony have a complex relationship in their own story. By listening to Tony fanfiction, he could be trying to gain a new perspective on Tony. It's like looking at a familiar painting from different angles. Fanfiction offers different takes on a character, and Steve might be curious about those interpretations.
It could mean to listen to a story until it's completely finished. For example, when you're listening to an audiobook or a story told by someone, you don't stop in the middle but keep listening till the very end.
The 'Tony Takitani' short story often explores themes of isolation and the impact of consumerism. Tony Takitani himself is a rather solitary figure. His life seems to be a continuous cycle of work and solitude.
The Hunter x Hunter manga did come to an end. However, its conclusion was somewhat controversial among fans, and some felt it was too abrupt or left certain plotlines unresolved.