Eric,

You aren't rambling. You are totally on the right track. A good book that covers the math on what you want is: "Solenoids, Electromagnets, and Electromagnetic Windings" by Charles N. Underhill. ISBN #1-55918-096-X. This book was first printed in 1914. I like the old books because they are factual, and uninfluenced by changes in theory and politics. It is currently available from Lindsay Publications as a reprint.

This book will let you calculate magnetic field strength for wire size, number of turns and current. Teal's big secret was the method whereby he minimized the air-gap in the solenoid at the middle of the stroke. This sort of solenoid design is not discussed in the book, so you will not be able to accurately calculate the mechanical energy produced. You will have to build a model of the machine and MEASURE the mechanical energy generated to fully understand what it does.

Peter