Keep in mind that I'm not an expert and I'm going with what is generally considered acceptable by the mainstream, both for what the
planets are like and the causality we've attached to these
characteristics. I don't necessarily agree with all these explanations
and correlations, but it corresponds to common understanding.
Based on what we know about planetary formation and geology, the mantle and liquid core provide us with a magnetic field. Without one, we'd have no protection from the solar wind:
That means the Sun would evaporate the atmosphere and hence you get less protection from solar and cosmic radiation. You also get less weather, such as lightning, rain etc. - without these, you don't get much chance for evolution of complex life, though of course extremophile bacteria can still survive. So you might have "life" in the strict sense, such as bacteria landing through space, but you wouldn't call the planet "alive" in any sense.
Liquid water is somewhat related. Considering that the primitive Earth is assumed to not have water the way we have it today, but a much more toxic mix that slowly got changed into a combination of mostly water, due to environmental change which came about from terraforming bacteria (this is how I remember it roughly, not sure on the details), I'd say we're back at square one - you won't have vast populations of microorganisms doing their business on a planet unshielded from solar radiation (especially when the temperature range would be much more than 5-10 degrees Celsius between night and day - it'd be closer to 100 or more, making any adaptation difficult). You could however have ices, accumulating from space debris over a long period of time. The extra problem here is, where would it collect?
The reason we have vast oceans is due to mantle activity - tectonics, volcanos etc. Without a liquid mantle, you don't have those as far as we can understand. So you'd just be a smooth rocky planet like Mars. No basins.
The last but not least problem I can think of is the evolutionary cycle. Major extinction events have been linked to the emergence of new and more complex life. In our case it was probably related to environmental changes, correlated with environmental weather and atmosphere cycles, as well as solar cycles. This requires enough stability to preserve life after a disaster, but not enough to prevent the cycle from moving on, to allow newer life to take over. In other words, even if you got to dinosaurs, they could just stay like that for much longer (assuming extinction events are not caused by biological populations and instabilities inherent to evolutionary stages).
A planet without an atmosphere, no mantle movement and eruptions and little shielding would be pretty much like Mars - rocky, barren, relatively smooth and practically lifeless.