1. As it's the nearest temple to the library there's probably something distinctive about it (a big dramatic tree out front, or an eye-catching archway, maybe), but Jay chose it simply because it was the closest one. Feel free to claim that something distinctive about the temple could be the reason Jay chose it (and to make up whatever that distinctive thing is!)

2. If Lavi goes through the shrine, he'll find a blood trail on the other side, in the Near Shore. The shrine in the Near Shore is a little one on a quiet street, but just a block over is the start of a busy downtown. Without a dog to follow the scent it'll be really easy to lose the trail after just a few blocks.