These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Fridge Logic: This show really runs on it. Sometimes it works, other times... not so much.
Fridge Brilliance: Agent Carter is so obsessed with avenging her dead partner that she can't put the pieces of Booth's existence together (which really shouldn't be much harder than comparing all known information on Booth's whereabouts with Gus's alibis. Booth left a trail of bodies across the country, and it's unlikely he just happened to do all his murders while Gus was on sabbatical from his professorship.
Booth set it up that way. He had been following Gus for quite a while. When Carter goes over the list of victims and when and where they were murdered, they fit where Gus would have been at the time.
Booth is trying to give Gus a sporting chance at clearing his name; if Booth wanted to just have Gus live his life as a fugitive, he could disappear somewhere Gus would never find him, or kill himself somewhere the body would never be recovered. By staying alive and staying in the Pacific Northwest (the region both brothers grew up in), Booth is giving his brother a real chance to prove his innocence.