Progressive settings look flickery on moving objects compared to interlaced settings. The slower the frame rate (24P), the more the flicker is evident. The advantage of progressive is that it has a more “filmic” look, and can look better when viewed on a computer monitor (which uses progressive scan). Also, frames are easier to work with in graphics programs, like After Effects.
Action footage is typically going to look better interlaced. I would consider using progressive for interviews and lock-down (tripod) shots where you want a filmic look.
Don’t try to deinterlace interlaced footage (by using it in a progressive project, for example) – it cuts the vertical resolution in half. On the other hand, progressive footage will be fine in interlaced projects, albeit with some flicker on moving objects (no different than it would look in a progressive project).
Higher shutter speeds reduce motion blur, and increase flicker, so they are counter-productive if you want less flicker.