The gist is correct, but it's worth noting that, for example, if you played do-re-mi in E it would do E-F#-G#, not E-F-G, to preserve tone/semitone order. That's where the key signature comes in.
It looks better on paper for an engineer, but not for someone who actually plays an instrument and reads the notation. With most western music, not all 12 pitches in an octave are used most of the time, but only a subset determined by the key and scale. Although the currently used notation may look weird for a newbie, it takes just a quick look at the key signature and you know which pitches will be used in a piece of music. When you know the scale (and practicing scales is just a standard part of learning), then "decoding" a note by counting tones is much easier than counting individual semitones (12 seems just too many). After a little practice you get it intuitively and you really don't count; you just know where each tone (or chord) is in a given scale and what function it has. And then when you suddenly see an additional flat or sharp symbol before a note, you know that this is an out-of-scale note, so it is also easier to play it. Disclaimer: I'm an engineer.
That completely misses the point. Without wanting to sound brash, you know jack about music and you should act accordingly, that is, don't spout ill informed suggestions when you clearly don't play an instrument or studied this mater well enough to give an informed opinion.
Scales have 7 notes, not 12. A musician plays music in a scale, they aren't a computer outputting pitches, they are a person playing notes. Music notation has a reason to be this way: notes in the scale don't have flats or sharps next to them, accidentals do. Reducing everything to a number describing absolute pitches is the right thing to do for a computer to play (see midi). It's not the way to go for a person that actually has to understand the logic and patterns in the music.
Db would be 01. Dbb would be 00. (or 11 and 10 for the octave)
The point ajuc is making is that the flat-sharp accidentals aren't used or needed at all if you just assign numbers to each tone. There's no concept of flat or sharp, unless you want to deal with microtones.
The letters and flats/sharps give you key/value over frequencies, which is better than just a numeric index over a chromatic scale. Working with the keys allows for the same abstractions to be used with all 12 keys at the same time, on the same staff.