How fast does arduino process each line of code?

Also say you have the most simplest photo flash type code of all, no delays, just a simple "when input #1 is high, make output #1 high". What sort of timeframe does it take for it to actually register the input and produce the output? Is it within the range of mere nanoseconds, or microseconds, or what?

Microseconds. You could probably get it down to the nanosecond range if you use an interrupt. Not really sure exactly how quickly the avr responds to an interrupt. Never had the need to time something anywhere near that fast.