You totally could pump it out, and I suspect that's what humanity will have to do, as clearly we are not going to stop pumping it in for a while...

Out of the ocean might make far more sense, as the energy costs are lower. There's exchange from atmosphere to oceans.

Several options: pump it back down the oil wells it came from, use fast-growing juvenile trees to "pump" it into house frames and libraries, use algae to "pump" it into useful plastics, or, if you have infinite money, bottle it and shoot it at Mars to thicken the atmosphere.

You also forgot synthesizing diamonds, which, pulling it out of my butt, I imagine is on similar energy budget with dumping it on Mars.

Solar sails and balloons wouldn't require infinite money. Merely a lot.

Ideally into solid carbon bricks that could be used as a building material and last practically forever.

Fix it to some stable form. Hydrocarbons, limestone, solid carbon ...