If using the word president in a sentence is it capital. Refering to our countries' presidents.

Answer 1

You only capitalize president when it is used as a proper noun, as in:
"Give this note to President Obama."
"Please respond to the question, Mr. President."

You don't capitalize it otherwise:
"Hilary Clinton is running for president."
"The president of the class is Timmy."
"One day even you can be president."

Answer 2

The rules given and the examples are all correct.