General Galactic emerges from stealth to make methane from carbon dioxide

They are working on the tech, not rolling it out.

To answer your questions:

1) Capture stations will be setup wherever the people who license the tech or purchase equipment want it to be setup.

2) The easiest way to guarantee the electricity used is Green is to only plug the equipment into Green electricity sources. Which makes economic sense too if you can afford the initial outlay. Plus it means you can sell your methane as a Green fuel without being put on trial for fraud, a bonus if that is a better market.

For hydrogen in Europe to be considered green - if I’m not mistaken - you need not just carbon neutral sources, but also you need them built specifically for the purpose of capturing hydrogen. It can be tracked.

The physics are very possible. The challenges of scaling and making it economically viable, now thats another matter.

Usually the proposal is massive solar arrays, and maybe only producing during daylight hours to avoid battery costs, and spreading the upfront build cost over the solar array's lifespan. I think a lot of this falls down though when doing the finance, as people realize they can make more money selling the electricity to the grid rather than using it for their energy intensive project.

With the current trends in the economics of energy, the thermodynamics is not really a limit: the electricity you need is generally available cheaply (or even at negative prices) at certain times of day. The biggest issue is getting anywhere close to that thermodynamic limit, and the cost of building the equipment in the first place. Capturing CO2 from air takes a lot of surface area and a lot of energy before you even do the 'turn into methane' step, and because the energy is only available at certain times, you can't run your plant continuously so the initial costs amortise badly. Those are the areas that need solving to make this concept work.