It is a convention that simplifies some formulas. You can always choose to include the propagators on the external lines, but their effect is rather trivial so they don't add relevant information to the diagram.
"One particle" means "one line". The reason for the word "particle" is that lines represent particles (sometimes, virtual particles). The reason for the word "one" is that the diagrams remains connected after cutting any one line. You can also have two-particle irreducible, if it remains connected after cutting any two lines, and more generally, $n$-particle irreducible if it remains connected after cutting any $n$-lines.
Remark: in $1.)$ I am assuming that you are using Feynman diagrams to calculate correlation functions. If you are using them to calculate $S$-matrix elements, then the external lines are amputated because of the LSZ theorem.