Regarding the count, why does the count start with a 2 ?

Historically (I think) Cha-Cha developed from Mambo, which breaks on the 2 beat. Musically (an instructor pointed out) there is a difference between starting a movement on the downbeat (1) and the upbeat (2). That's the reason why Mambo and Salsa look different even though many of the steps are the same. Breaking on the downbeat makes it look more "laid back." Breaking on the upbeat makes it look "sharper" (more "upbeat").

Technically, the rhythm (slows and quicks) is the same but the timing (which beat of the music a movement starts on) is different. The instructor I mentioned demonstrated this by dancing a simple Waltz box, but stepping forward on the 2 beat instead of the 1 beat. And he was right; it looked different.

You can start with 1. It's called a prepatory step. Rumba is the same way, where in international style you do a prep step on 4-1, and start on beat 2. American style rumba starts on beat 1. Cha cha is the same in both.