On an industrial scale, yes, of course.

Yes

Yes, by burning fuels we release carbon from the thing being burned, it gets mixed with oxygen, and released into the atmosphere as co2.

As the suns energy warms the earth, the earth radiates heat back out and away from the surface and normally back out into space. co2 also acts like a blanket for that radiant heat, and traps it inside.

More co2 in the atmosphere traps more energy around the planet like an insulated wrap.

To reduce this, we have to extract the carbon out of the atmosphere and store it.

It's the main greenhouse gas contributing to climate change. Where have you been?

Sure, but it's only damaging globally. It's no more damaging to the local environment than to anywhere else on the planet.