What if we're not the first AI-capable civilization on Earth?

Well, no.  That's not an interesting question.  Whether or not we're not the first such civilization, we STILL experience everything exactly the same.  Reality is truth.  We can't change the past and see what's different.  There's no "what if" that makes sense for that topic.

Perhaps my choice of the phrasing was sub-optimal (I'm a non-native English speaker). The intent of this question-post is to identity interesting corollaries of the hypothesis (including testable predictions).  

You don't give any reasons to give it serious consideration.

The core justification for this hypothesis is that it's basically SETI, but for civilizations in the Earth's past instead of civilizations in the outer space. And many people in the community seem to find SETI worth a serious consideration. 

 It's quite possible that such local and brief civilizations have happened and collapsed multiple times.  I would still ask "so what" before "what if".

Regarding "so what", I don't know yet (a purpose of this question-post is to explore this kind of stuff). So far I can see two interesting corollaries:

• we can apply something like the Drake equitation or "grabby aliens" ideas, but to the hypothetical previous civilizations of the Earth. This could give us some insight about our own future
• the possibility of the ancient genocidal AGIs still roaming around (or lying dormant in some ruins) - seems important.

Among dinosaur candidates, Troodon seems to be the most interesting:

• a bipedal species, with agile hands and stereoscopic vision [1]
• the structure of the teeth suggests a preference for soft food [2] 
• a large brain for the animal's size [1]
• social (it's known that their nests were shared by multiple females) [4]
• massive concentrations of atmospheric CO2 during the time (higher than today) [3], followed by one of the largest extinction events in the Earth's history (the Cretaceous–Paleogene extinction event)

That seems unlikely - our genetic diversity provides evidence of how many humans there were at any point in the past. We would notice if there were billions of humans only a couple of hundreds of thousand years ago

Judging by the existence of the highly sophisticated Ancient Greek civilization (capable of devising and making the Antikythera mechanism, a mechanical computer), there is no necessity for a technological civilization to be billions of people strong. At the time, the entire population of Europe was about 34 million, comparable to the today's population of California. 

Perhaps the hypothetical previous civilization was not a global behemoth like ours, but more like the Ancient Greece.

By the way, it took Greeks only 7k years from the first stone-age agricultural societies to the Antikythera mechanism (and also to simple steam engines and all kinds of other interesting tech). 

I wounder how many such brief sparkles of civilization the Earth has seen, with the ruins buried under kilometers of sediment, or vanished without any trace. 

I'm guessing that we would still be able to notice that in the distribution of minerals - e.g. maybe all the high iron concentration ores should have been mined, but I don't really know much about this.

Troodonts became extinct about 76 million years ago. Since then, even some continental plates have been swallowed by the Earth's crust. But I agree, perhaps it's possible to infer a possible advanced civilization using geological or paleontological indicators. E.g. certain patterns of mass extinctions, an anomalous rise of CO2 etc.