I believe it's
It tells us the winner when equines do race.
Horse races can be won "by a head."
To my misfortune there's one on my face.
Referring to acne; i.e., a whitehead or blackhead
Our countries have many, some say full of gas.
Heads of state are often not well-respected.
They could also be considered airheads. (Thanks to @feelinferrety.)
My sister was one when completing her class.
The head of the class is the top-performing student.
Sometimes it's level, sometimes it's not.
People can be level-headed, sensible, and can also not be.
The coin in my hand, just one has it got.
A coin often has one tail, and one head.
Up at the top it starts a great flow.
The head a river is its source.
The one on my face is ready to blow.
Oh, I just got the title: Figure this out while I relieve myself
Head is another term for toilet, where one can relieve oneself.