1."informally to argue your case in front of other people.."
Could you please explain what this means?
I will give you an example.
Suppose that your brother is accusing you of taking his favorite book. He found his book in your room. You tell your brother "I didn't take your book.", but he does not believe you. How can you make your brother believe you? You find a witness.
Your sister witnessed (saw) your brother's friend taking the book, looking at it, and then putting it in your room. Your sister is your witness. She will tell your brother that you didn't take the book. She will argue that you are innocent.
2. In the example, it is not formal.
3. Birders are people who have a hobby of watching birds.. They try to find rare birds and identify them. They carry binoculars and books that have pictures of different birds. They get up before sunrise and travel to places where birds might be. Owls are especially hard to spot because they fly and hunt at night and sleep in the day time.
The sentence says that the birders were successful in spotting (seeing) the Great Horned Owl (a kind of owl). "Spot" means that they did see it, but only for a short amount of time.
4. The police saw the robbers later on that day.
If I use 'saw' here, does it have the sense of looking for the robbers before seeing them?
No, not as much as "spot". "Spot" has the meaning that the police were looking for the robbers, and saw them. "Spot" also has the meaning that it was a surprise (they didn't expect to see them), and they didn't see them for very long.
5. Yes.
"Spot" also has a meaning of surprise, such as seeing a very famous person when you didn't expect them. Here is a real example:
Brad Pitt was spotted leaving the seafront Lenval hospital Wednesday morning with his children Shiloh, 2, and Zahara, 3.