That is exactly the issue. If you really want to solidify the carbon for long term storage, you'd have to spend more energy than you gain in the first place. At least I'm not aware of a process that could capture CO2 for long term storage with only a fraction of the Carbon-to-Carbon binding energy (which is what you gain when burning, minus the C-O2 binding.

This is exactly what photosynthesis accomplishes, using energy from the sun.

At which point you can just use solar cells and batteries? I mean you do get high energy density with coal, but the downsides just seem too much.

Isn't liquid CO2 used as a coolant? Sounds like we have our global warming solution.

Only if by "Global Warming Solution" you mean that it will cause more global warming.

Liquid CO2 boils away (solid CO2, i.e. "dry ice" sublimates) as it cools, so it'll still be released into the atmosphere.