I am using the Beaglebone black rev. A6A.
What is the state of the GPIO pins during boot up? I am using a real time
clock to turn on and off the BBB. Is there a problem connecting a BBB's
GPIO line directly to an output of a device that is already on?
Depends on the I/O line. Almost all of the LCD_whatever pins are used to boot up. In addition there are a few I/O that are driven by default until the overlay is applied and the pin is exported for GPIO.
I'm not sure on the specifics but I have always run under the impression that power applied to an input with a device that is not connected to power is to avoided.