

It probably would get her in some hot water, but a first name alone is a gray area. It becomes a definite violation if it’s combined with health information, even as simple “a baby was born here with the name X”. If she just says “I saw a name spelled X” then it may not be a violation of the law, but the hospital would probably still can her for it.
The process you’re thinking of is oxygenation, not oxidation. Oxygenation is the binding of oxygen to other molecules, oxidation is the loss of electrons. When the iron in hemoglobin oxidizes (from Fe2+ to Fe3+) it stops binding with oxygen, and if it oxidizes further (to Fe4+) it can start oxidizing other molecules in your body. Your body has enzymes to reduce the iron back to a reactive state, but antioxidants also play a role in reducing oxidized molecules.