The reason we avoid E# and B# is to get nice-sounding chords by only using the white keys. This way, the C-E chord has a ratio of 2^(4/12) which is approximately 5/4; the C-F chord has a ratio of 2^(5/12) which is approximately 4/3; and the C-G chord has a ratio of 2^(7/12) which is approximately 3/2.

In fact, before we understood twelfth roots, people used to tune pianos so that the ratios above were exactly 5/4, 4/3, and 3/2. This made different scales sound different. For instance, the C major triad might have notes in the ratios 4:5:6, while a D major triad might have different ratios, close to the above but slightly off.

There's also the question of whether the difference between these makes a difference in the sound. There's two answers to that. On the one hand, it's a standard textbook exercise that the difference between pitches of a note in two different tuning systems is never large enough for the human ear to hear it. So, most of the time, the tuning systems are impossible to distinguish.

On the other hand, there are certain cases in which the human ear can detect very very small differences when a chord is played. To give a simple (though unmusical) example, suppose we played a chord of a 200 Hz note and a 201 Hz note. The human ear, to a first approximation, will hear a single note of approximately 200 Hz. However, the difference between the two notes has a period of 1 second, so what the human ear actually hears is a 200 Hz note whose (EDIT) amplitude wobbles every second. This is very very obvious, it's a first sign of your piano being out of tune, and in different tuning systems it happens to different chords.

Notes sound good if they're approximately simple rational multiples of each other. Hence you want your scale to contain multiples.

Since the simplest multiple is x2 we use that for the octave. As for why we break it up into 12 semitones, the reason is that 2^(7/12) is approximately 3/2 and as a bonus 2^(4/2) is a passable approximation to 5/4.