I know how the theory works in a jar.
Is the +0.5C just the troposphere or the entire atmosphere as a whole?
The +0.5 is supposed to be just the troposphere. It's essetnially a surface / near-surface reaction. Shortwave radiation from the Sun hits the planet's surface where it turns to longwave radaition (given its properties as a partial blackbody). The longwave radiation would then careen off into space, no problem. This is why the moon cools down so much at night. There's no atmosphere and all that longwave radaition goes off immediately into space. However on Earth, it doesn't go down to -200 at night.
The theory is that greenhouse gases, notably water but also CO2, absorbs the longwave radiation that's heading out, and redirects about half of it back to the surface of the earth. Where it is subsequently reradiated. This delays things near the surface which warms up the surface but results in cooling in the upper levels (since you can't just "create" heat from nothing, you have to have cooling elsewhere to balance).
From what I gather we've measured an increase in tropospheric temperatures, and we've measured a cooling in the stratosphere.
The theory is that if you add CO2, you increase the amount of spacebound longwave radiation refelcted back to Earth. About 1 deg C for every doubling of CO2 (since it's a logarithmic relationship).
I know how the theory works in a jar.
Is the +0.5C just the troposphere or the entire atmosphere as a whole?
The theory doesn't work in a jar. That's part of the problem. As soon as you stick it into a jar, you've put the gas in a vessel that is transparent to optical light but more opaque to infrared. Also, a jar doesn't let convection happen.