Are there tell-tale signs that clay will crack when fired? I understand there are many different kinds of clay, so maybe you could offer advice for clays that will be typical to beginners, or other otherwise buying from common arts stores.

I imagine that red and grey clays are different in this regard, but maybe not?  I know that bubbles are an obvious issue, as is insufficient mixing of add-ins, and large inclusions.
Yes. Look out for "veins" in the clay as it is altered and manipulated from its original resting state.

Veins might also look like threads or mini cracks.