To answer your second question, it depends what you mean by an isomorphism. In general an isomorphism is a bijection which preserves all functions and relations in your language.
So an isomorphism of SETS is simply a bijection because there is no other structure.
As a less trivial example, an isomorphism of ORDERED sets is a bijection which preserves the order.
So as sets, the naturals and the integers are isomorphic. But as ordered sets (with their natural ordering) they are not.
These are just 2 examples. A structure can have many relations and/or functions that need to be preserved.
For your first question, the reals are the unique completion of the rationals. Since any countable dense linearly ordered set without endpoints is isomorphic to the rationals, then the reals are the unique completion of any countable dense linearly ordered set without endpoints (up to isomorphism). By isomorphism here, we mean an isomorphism of ordered sets.
Here is a reference: "Introduction to set theory," by Hrbacek and Jech, chapter 4.5.
This material is pretty standard however, and can be found in many places.