Short answers, pulled from the story.
George Beauchamp and Paul Barth designed the Ro-Pat-In Electro A-25 known as the Frying Pan. They served as general manager and vice president of the National Stringed Instrument Corporation respectively when they introduced the instrument in 1932.
The first public performance using the Frying Pan took place in Wichita, Kansas in October 1932. Gage Brewer performed this event which demonstrated that the electric guitar could be a viable instrument for live performance.
The first mass-produced solid-body guitar was the Fender Esquire and Fender Broadcaster which later became the Fender Telecaster. These models were first made in 1950 by the Fender company.
Double-coil or humbucker pickups were invented to reduce or counter the sound of mains hum. They are designed to buck the hum by using two coils with high combined inductance to create a richer and fatter tone.
A typical Fender guitar uses a 25.5-inch scale length while Gibson uses a 24.75-inch scale length in their Les Paul. The scale length is the vibrating length of the strings from nut to bridge and affects fret spacing and tone.