Apple - What operating system was installed on the original Apple 1?
According to Mactracker, it was "Apple Integer BASIC with optional Cassette Interface".
I used to own an Apple ][. It had floating-point BASIC installed in the upper 16kb of memory. "Booting" took maybe 1 second - flip the switch -> Beep -> it's ready.
If you bought a disk drive, the boot loader was in a 256 byte (bytes, singles, no M or k) ROM on the interface card. Apple DOS was written onto every floppy disk and took up about 10kb of RAM.
Some trivia for your project:
The Apple ]['s documentation included a fold-out circuit diagram of the computer's electronics. Just in case you needed to repair it.
The documentation also included the complete source code (6502 Assembler) of the operating system. Maybe 30 pages. Assembler at the time was a 1:1 representation of the machine code.
I wrote Apple (on paper - no internet back then) asking for the Integer BASIC source code for comparison. They wrote back and said they would be happy to send it, but they don't have a copy. Wozniak apparently wrote Integer BASIC himself, compiled it with a pencil and the Motorola 6502 data sheet, and typed in the resulting hexadecimal opcodes by hand. All bow to the Greatness of The Woz.
The original Apple 1 came with a 256 byte ROM containing Steve Wozniak's monitor program. This program allowed the user to inspect memory, write to memory and run programs. There was also a very basic line editor that permitted the use of backspace and escape keys.
The Woz Monitor
Apple 1 emulator in Javascript
If you want to try out the emulator, be aware that the Apple 1 does not have a power-on reset and requires the user to hit the ESC key to initialize the monitor.