2. To find the direction of $\vec{u}\times\vec{v}$, hold your right hand so that your index finger is pointing in the direction of $\vec{u}$ and your other fingers curl toward $\vec{v}$, the your thumb will be pointing in the direction of $\vec{u}\times\vec{v}$. Basically that just tells you that $\vec{u}\times\vec{v}$ is perpendicular to both $\vec{u}$ and $\vec{v}$ and $\vec{u}\times\vec{v}= -\vec{v}\times\vec{u}$.
$\vec{b}\times\vec{a}$ is perpendicular to both $\vec{a}$ and $\vec{b}$ $\vec{a}\times (\vec{a}\times\vec{b})$ is perpendicular to $\vec{a}$ and $\vec{a}\times\vec{b}$ but not $\vec{b}$. To see that more clearly, take a specific example: if $\vec{b}= \vec{i}$ and $\vec{a}= \vec{j}$, then $\vec{b}\times\vec{\a}= \vec{i}\times\vec{j}= \vec{k}$ so that $\vec{a}\times(\vec{b}\times\vec{a})= \vec{j}\times\vec{k}= \vec{i}= \vec{b}$.