Move featured images into header area with post title overlay

[ Moderator note: moved to Fixing WordPress. ]

I am using a child theme. I want no featured images displayed in their current positions. I am even deleting their appearances on the frontpage. Instead, here is how I need the featured images to appear:

I want the featured images attached to articles in the header, where the regular site header images normally go. This space should be the same size as the frontpage site header area.

This is where I imagine it gets complicated:
Because not all my writers remember to attach a featured image, I want any story without an attached featured image to fall back to a specified default image (or even a set of images).

To be more in line with modern design, I need the post title to appear on top of the header image, overlaid like an image macro (or meme).