Can a planet have a 6 Earth-month rotation around it's star at the same time it has a 12 Earth-hour day?

Yes. You'd need to significantly increase Earth's orbital speed around the Sun, while simultaneously reducing it's angular momentum.

How does this work

Not too easily, since putting retro-rockets on rotating masses the size of Earth is really difficult.

is it possible?

I doubt whether even a Dyson-sphere civilization could do it.

How does this affect the climate

Being much closer to the Sun, with less time each day (1) exposed to the Sun and (2) cooling off would naturally have a severe impact on the planet. Probably would boil off all the water.

No, it wouldn't be habitable.