That's simply due to the nature of evolution. It's nearly impossible to look at one past generation of chicken to the next to figure out when the ancestor was no longer a chicken. Yet, go back far enough and you'll find T-Rexes in the mix.
Every generation is a new missing link. It's an extremely fuzzy process.
Afaik, T-Rex was never a direct ancestor of modern birds, including chicken. T-Rex and birds are theropod dinosaurs, but it was a very large and diverse group of animals.
It depends on what you mean by the age of the species. You can find the oldest known fossil occurrence at the Paleobiology Database [1] and the divergence time from molecular phylogenies via TimeTree [2].
The number is both incalculable and vague - is a shark tooth enough to count as a fossil? How about diatoms and other microfossils?
Diatomaceous earth alone contains around 10^6-10^7 frustules (the shell of a diatom) per gram. If you count them as fossils then the lower bound is 10^18 fossils per year just in diatomaceous earth production (the fossils are ancient but we produce nearly a million tons a year in diatomaceous earth).
Then consider what's buried under the sea, totally inaccessible. Or under the ice at the poles.
It's a lot of fossils. And that's without even getting into questions like "what counts as a fossil for these purposes?", just any halfway sensible answer is going to leave you with an unfathomably big number, no need to even dig (ha, ha) into the specifics.
The places scientists go to dig up fossils are mostly where a particular stratum happens to exist (the crust gets recycled, so much of the oldest stuff is simply gone in most of the world) and happens to be exposed near the surface. Those same kinds of (for the more common strata, anyway) exist all over the place, just buried too deep to get at except, sometimes, during commercial excavation for things like mining (and then most of it's just gonna be destroyed without a look).