Answers are presented in bold with a brief explanation following.
1. RELATING TO EXPLORATION IN THE 1500s
Cultures with different religions and views about the origin of the world were encountered.
It is important to note that it was the different views about the origin of the world that led to questioning the dominant ideas in Europe. It was not simply "new things" but the kinds of new things that made an impact in Europe.
Many new species were encountered making it impossible to remember all their names and necessitating creation of classifications to form larger groups that could be remembered.
Again, learning about new species was exciting, but it was the necessity for an easy taxonomy (naming system) that prompted naturalists to think more closely about how species were related (in a physical sense, not a genetic sense).
Additional species that had sets of relatives elsewhere increased the impression that there were minor differences among species, suggesting that species were similar because of common ancestry.
The important thing is that there are minor differences between the species and its relatives, but that they are nevertheless distinct. This observation contrasted with the classical idea that each species had an essence or type that exemplified that species.
Patterns of geographical distribution among living species indicated that species in each place had not been specially created but that similar species had a common ancestor in the region.
This is similar to the previous answer.
Species of modern and fossil organisms in the same region were more similar to each other than to species in other places, suggesting that the fossils were ancestral to the modern forms and differences between them were due to evolution.
The emphasis here is on the close similarity of local fossils. See the responses to question 2.
It became less and less tenable to propose that extinct species would be found elsewhere in the world and that there extinction was impossible, as suggested by biblical scholars.
As more of the world was discovered, the chance of discovering living examples of extinct species decreased. The observation that species go extinct was quite contrary to the dominant view supported by the church.
2. RELATING TO GEOLOGY IN THE 1700s
Uniformatarianism -- mechanisms that can be observed in the present caused phenomena that occurred in the past.
This idea prompted naturalists to find a uniformitarian way of explaining where life came from.
Gradualism -- mechanisms that take place slowly can produce major changes (differences) if they are applied over long periods of time.
Again, naturalists could wonder if new species could arise gradually.
The world is much older than the biblical interpretation indicates, old enough to give a slow process like natural selection enough time to produce major differences between distantly related species.
The importance of this discovery was that it made it possible for natural selection to work.
Many species that existed in the past are extinct now.
A simple discovery, but an important one. If extinct animals are no longer here, where did they go? What happened to their offspring? Did they just die, or are their many-times-great-grandchildren still here somewhere?
Species that look like ancestors tend to occur in the fossil record earlier than those that appear to be their descendants.
This is another way of saying that things appear to be in a continuous order in the fossil record.
Species in the fossil record are more similar to modern species as one goes from older to younger rocks.
The importance here is the continuity of the fossils.