Uh, Flaser, I accounted for that. There are only a limited amount of radioactive elements in the planet's core, and eventually, the bulk will no longer give up energy to heat the planet. The planet's core will cool into a rock.
Though I am curious where you got the idea that the core would "suck heat" from the planet's surface. The Earth puts heat out via vulcanism, the core is, in this way, constantly cooling. Even with all the radioactivity going on, the rate of cooling is increasing. The planet's climate, however, isn't caused by vulcanism in the long term (when the planet initially forms it is important, but not after a few billion years), it's caused by the sun and solar radiation. Even if the core suddenly froze, the enviroment of the Earth wouldn't change too much as it is the sun that keeps us warm, not the Earth.
The early atmosphere of all planets is probably generally the same, CO2, water, various other gasses. Water, in the form of water vapor, enters the atmosphere just like anything else. Eventually, it will condense (assuming proper conditions) and fall to the surface as rain, eventually forming streams and oceans. Which leads to life unless the planet isn't in just the right place. When it's right, the planet looks like Earth, when it's wrong, it ends up like Venus or Mars.
Getting a Vasuda Prime, that's a lot harder. You need to, somehow, remove water from the planet (assuming it starts with the same amount of internal water as Earth and such, which is reasonable). The best way is to heat the planet up, and boil the water into the upper atmosphere where solar radiation breaks it up and blows it away. The other way is to say the planet doesn't start with a lot of water in the first place, which is just as valid.
In any case, if the plant is still geologically active, with lots of volcaneos and the like, more water vapor will be introduced into the system, allowing the planet to reclaim what it loses. Since maintaining the level doesn't make sense given the eventual result, a desert world. So vulcanism must all but stop. This means the core has cooled so much that the heat is no longer sufficant to even break through the thin crust. The planet is dead, geologically speaking.
As for Mars, the reason it's not inhabited by Martians isn't because of it's amount of greenhouse gasses, it's atmosphere is something like 90% CO2, it's because the atmosphere is so damn thin! The result of the planet being very small, and atmosphere blowing away in the solar wind. The atmosphere needs to be thick, first and foremost, which means the planet must have a significant amount of gravity. At a minimum, as I think we can all agree, Vasuda Prime is as large (mass wise) as Earth, perhaps bigger.
My idea makes Vasuda Prime both old enough for the planet to geologically die, and warm enough to become a dry, desert world. All that has to happen is to wait. The habitation zone around the star, where water is in a liquid state, moves outward as the star ages. Any planet within said zone will, eventually, leave it, either by being eaten by the star in it's death throws, or simple via time. I think Vasuda Prime, regardless of where exactly it sits, probably is nearing the exit point, or is there already.
This heats the planet, forcing the water to vaporize and be transported to the higher levels of the atmosphere where the stronger solar radiation can do it's work and boil it away. The planet eventually begins to dry out. The water cycle is broken, leaving only bits and pieces of a water table scattered about and an extremely salty ocean. There is no rain, no percipatation of any kind. Just a dry, desert world, a dying world on the edge of oblivian.
Sorry for misspellings, I've got a cold or something and am sleepy. Later.
