When you get your new one besides the application you'll need to fill out an additional form; DS-64 "Statement Regarding Lost or Stolen Passport". When you get your new one in one of the pages at the back will be a small typing that says "This Passport is a replacement for a lost or stolen passport".

I haven't had any issues with my replacement one, and certainly haven't been flagged every time I use it. I've never experienced the visa issue, but I imagine you'll need to get an entirely new one.