Perhaps a chemist can help answer a question that this raises. These scientists were careful to use formaldehyde because it has been detected in interstellar space. But then they treat it with lime. Normally, lime is created from limestone. Isn't limestone created by the actions of living things such as shellfish? Does lime occur naturally, too? If so, is it stable enough to remain in that form, or would it rapidly turn into some other compound? If lime does not occur naturally, are they postulating that it may have occurred naturally 3 billion years ago? Otherwise, aren't they relying on the occurrance of a byproduct of life to explain the emergence of life?
Yes, whether you mean unslaked lime (CaO) or slaked lime (CaCO3). CaO is less stable (strongly alkaline) and will form CaO3 with CO2 in the atmosphere.
Most limestones are of marine fossil origin, but not all calcium carbonate minerals are such classic limestone. There are non-fossil forms such as aragonite. Even in the fossil cases, the marine life that secreted the shells in the first place got the calcium from dissolved calcium salts in the ocean just the way they do now. Both forms of lime are unstable in the presence of hydrogen ions (that is, even weak acids). That's why there are lots of caves in limestone formations.
I don't know how reasonable it is to assume a specific calcium compound on the early Earth, but there were certainly calcium salts around then as now.