What is the difference between the letter 'O' and the number '0' (zero)?
In ASL, the letter 'O' and the number '0' use the exact same handshape. The meaning is easily understood based on the context of what you are signing, such as whether you are spelling a word or giving a phone number.
Does my palm have to face outward?
Yes, for the standard ASL manual alphabet, your palm should face outward toward the person reading your fingerspelling. This makes the letter easiest to read.
How tight should the 'O' shape be?
Your fingertips should gently touch the tip of your thumb. You do not need to squeeze them together tightly or tense your hand. Just ensure there is no gap, so it doesn't get confused with the letter 'C'.