To answer your question directly, it is quite unlikely that planets would be habitable without already having life. It would have to be in a near-perfect condition, which would be statistically unlikely. In case of planets already having life, the existence of life provides robustness to small changes and makes habitability possible in a larger range of conditions.
I am assuming 'habitable' here means habitable for man, trees and an ecosystem etc. Remember that there are extremophiles (or certain other hardy species) which completely change the perspective on habitability, and are often the key in making those places habitable for other forms of life.
This is also true for parts of earth which undergo a sudden change and are not 'habitable' for most forms of life till some hardy species takes over and changes the conditions, and consequently allows other species to be able to survive. See, for example, ecological succession (https://en.wikipedia.org/wiki/Ecological_succession), esp. primary succession and pioneer species (follow the links within the article).
Also, this is a long process. In case of the earth, there was also evolution that needed to happen, so it probably won't take as long as a few billion years, but even on earth, it takes a few years to make small disturbed microhabitats habitable. The atmosphere and other factors (soil, moisture etc., for the plants) need to be made suitable for living. Check this interesting article on terraforming: https://microbewiki.kenyon.edu/index.php/Terraforming
Hope this was helpful. I don't really know if a carbon cycle is possible outside of living organisms in some way.