In basketball, during a free throw attempt what side of the court should the official stand on
who is located at the top of the key? Should he be facing the score board or the score
table?

HS mechanics dictate that he (or she) face the scorer's table. There's no point in signalling to the scoreboard how many shots are left or when to start the clock.

Besides, if you were supposed to face the scoreboard and it was directly overhead, would you lay on your back?

2 or 3?
 

In 2 man the Trail official is opposite the table.
In 3 man the Trail is on the same side as the table, but there is a Center official opposite the table at the free throw line extended.