Is there an atmospheric hydrogen cycle?
"Is there any interaction between the Earth's atmosphere and high velocity protons in the solar wind (e.g. protons combining with atmospheric oxygen to form hydroxide) which is capable of balancing the hydrogen loss?" I found Hydrogen ions in the solar wind react with oxygen in interplanetary dust grains to form little packets of water, potentially supplying Earth with its water, but they don't know how much.

"How big is the hydrogen loss?" 3 kilograms per second.

"Is the amount of hydrogen on Earth in equilibrium?" We are also gaining 50,000 tons of matter per year from space dust, but only a small part of that is hydrogen.