Hydrogen escape from our atmosphere is diffusion limited, so this isn't a concern.

https://en.wikipedia.org/wiki/Diffusion-limited_escape

Hydrogen released into the atmosphere will mostly be turned to water by oxidation with OH radicals before it can escape. This is not entirely a good thing, as an OH radical consumed by reaction with hydrogen will not be available to react with methane -- and we want that methane scrubbed out of the atmosphere as quickly as possible.

Some quick research indicates that while we won't lose our water per se, there are potentially other issues along that line of thinking: https://www.caltech.edu/about/news/hydrogen-economy-might-im... (2003)

Edit: you might be thinking of helium, which is inert, and so I believe does escape the atmosphere more easily.

Hydrogen escapes the atmosphere because it is extremely light and the planet's gravity cannot hold on to it.

That just means you've invented a new fancy way to combat rising oceans!

(No, in reality, we have nowhere near the power to make appreciable amounts of hydrogen escape. Even if you electrolyzed water continuously with 2.5 TW of power, which is roughly global average electricity generation, you'd lose something like 4 cubic kilometers of water per year. The Earth meanwhile has over a billion cubic kilometers of water in its oceans, so you'd need over 250 million years to make them disappear.)

Only if the hydrogen is very high up. Hydrogen released at ground level will almost entirely be oxidized before it can get that high.

Yes Helium is the other gas that does this too - but Hydrogen is lighter than Helium.

Batteries vary a lot. Lithium iron phosphate seems pretty reasonable to me: they're made of lithium, iron, copper, aluminum, plastic, fertilizer, carbon, and probably a few more things I'm not aware of. Nothing particularly toxic or hard to recycle, and they tend to have very good longevity.

https://en.wikipedia.org/wiki/Lithium_iron_phosphate_battery

Escaped hydrogen will probably turn to water fairly rapidly. And by that I mean it will explode.

Depends on the environment that you're in. In an enclosed space: yeah. In open air, unlikely, because the density difference means it rises (and disperses) quickly.

In open air the halflife of hydrogen to oxidation is just a few years, which is much faster than the timescale for hydrogen to be convected up to the homopause.