That's true. It's the same. I believe setting these "()" is not strictly necessary, but it makes it better readible I guess

That's a good question. Actually I'm not sure: Normally we keep the notation $\displaystyle [a,b]$ as an interval in $\displaystyle \mathbb{Q}$, or $\displaystyle \mathbb{R}$.

Anyway: I think just writing $\displaystyle \left\{\left\{0,1,\cdots ,n\right\}: n\in \mathbb{N}\right\}$ Is suggestive enough in itself. Every mathematician will know what you mean ;p

$\displaystyle \left\{u\in \mathcal{P}(\mathbb{N}): (x\in u) \wedge (y< x)\rightarrow y\in u \right\}$ is a more formal notation.