Is the opportunity cost under the supply curve? Is it correct to say that the opportunity cost is a particular point on the supply curve? And if so, why? 

I understand what is going on with the opportunity cost with the production possibilities curve (how you give up more of one good when producing another), but I'm just not sure how it relates to the supply and demand curve, and area underneath the supply curve.

On this video, (https://www.youtube.com/watch?v=zEE-qOuo_Cw), at 3:15, he says that there is an opportunity cost more than $9. Why is this referred to as opportunity cost?
Opportunity costs occur for all suppliers. The lowest price at which supply occurs (agents willing to sell) is just above the lowest opportunity cost of the suppliers. 

I would not describe it as a point on the supply curve. Because you will not offer something for sale unless it is greater than you opportunity cost. It is important to note that opportunity costs dictate the starting point and slope of the supply curve.

So presuming your opportunity cost for 1 unit is 10\$ you will only sell at 11\$ (opportunity cost + 1), another persons opportunity cost may be 12\$ and so they are only willing to sell at 13\$ hence the increase in quantity supplied as price rises, more and more people are willing to sell as their opportunity costs are covered and they can make a profit.