TRUE, TRUE, and FALSE
The key to understanding special relativity is to realize that lengths and time intervals are different in different inertial frames, something that does not happen in Newtonian mechanics.

On p. 926 of your textbook, the authors say "The term proper is unfortunate in that it implies that any other measurement is improper or nonreal. That is just not so." I prefer to argue that proper time is a GOOD choice of terminology as long as we remember that the term proper comes from the french propre temps meaning "own time". Consider the example where event 1 is the start of an interstellar voyage by an astronaut, and event 2 is the astronaut's death from old age. The proper time interval between these two events must be measured by the astronaut's OWN TIME, i.e., by the astronaut's on-board physician consulting her own watch and calendar. Somebody back on earth might be able to measure that time interval also (using radio communications, and allowing for the transit time of the waves), but the time interval in a different frame is always longer than the proper time, according to the time dilation formula.