Short answers, pulled from the story.
A brass instrument is defined by the method of sound production, not the material. Sound is created when the player's lips vibrate against a mouthpiece, producing sympathetic vibration of air in the tubular resonator. This means instruments made of wood or animal horn, such as the alphorn or shofar, are classified as brass instruments.
Cylindrical bore instruments, such as the trumpet and trombone, have approximately constant-diameter tubing and are generally perceived as brighter and more penetrating in tone. Conical bore instruments, such as the euphonium, tuba, and flugelhorn, have tubing that increases in diameter throughout its length and are generally heard as more mellow.
Each valve adds a fixed length of tubing to lower pitch, but pitch perception is logarithmic while the physical effect of added tubing is inversely proportional to total tube length. This means no fixed combination of valve tube lengths can produce perfectly in-tune pitches for every valve combination. Players must compensate using triggers, throws, or lip-and-breath control.
The first piston valve instruments were developed just after the start of the 19th century. Heinrich Stölzel invented an early variety, the Stölzel valve, in 1814. By 1864, when Arban's method was published, a core three-valve layout had become almost universal.
Whole-tube instruments, such as the tuba and euphonium, have larger bores relative to their tubing length and can produce the fundamental pedal tone easily. Half-tube instruments, such as the trumpet and horn, have smaller bores and cannot easily play the fundamental, so the first overtone is the lowest practically usable note.
A British brass band typically includes one soprano cornet, nine cornets, one flugelhorn, three tenor horns, two baritone horns, two tenor trombones, one bass trombone, two euphoniums, two E-flat tubas, and two B-flat tubas. The ensemble is made up entirely of brass instruments, mostly conical bore.