Wow, you did a lot of work and got your machine opened up. I do some work on my machines, but you are BRAVE to delve so deeply into your machine. What make/model of machine is it? Maybe you can find the schematics somewhere online.
I bet you can get the tiny screws at your sewing machine repair shop, or maybe a hobby store? Hope you the best in repairing your machine.
I can't tell if it's true for your machine, but on some, when you have it set on a straight stitch but have the width set above zero, it sews a zigzag stitch. You might see if changing the width to zero makes a straight stitch. HTH
Well, the newspaper would make it a "foundation pieced block" and the irregular sized/shaped pieces make it a scrap quilt, and the shape would make it a star quilt. Here's a website with lots of scrap quilts, maybe you can find something similar there: http://www.quiltville.com/