A few questions about Abraham Lincoln's "stovepipe" hat

Was the popularity of this iconic hat on the upswing, or declining, in the 1860s? Was it widely worn by accomplished men of the day?

Would Lincoln have worn it to most official functions? Had it become such a personal signature that he felt compelled to wear such a visible target on Civil War battlefields?

Was Lincoln’s hat proportioned to his unusual height?

What is the history of this particular style of hat?