Timer1 has an Input Capture Trigger Source (ICP1, PORTB pin 0). This is a pin that saves the current timer value into ICR1H and ICR1L. You can then use the corresponding Input Capture Interrupt service routine to save the timer value and grab a byte of input bits to tell you which event occurred. I think the only external hardware you would need is a six-input OR gate so that all the inputs can connect to the Input Capture pin and a set of latches so that you can keep track of which inputs have occurred.
This should get you 62.5 nanosecond resolution.