Square root and principal square root confusion

For question 1, you are correct that you could factor $12=-3 \cdot -4$, but it is not useful. You are choosing to use $12=3 \cdot 4$ because that is useful. You know $3,4 \gt 0$ because you chose them that way. $x$ is a variable, and we don't know its sign, so we need the absolute value signs. We pull out the $2$ because we like $2 \sqrt 3$ better than $\sqrt {12}$. It is a matter of taste, and in some intermediate calculations we would prefer $\sqrt {12}$ or some other form.

For question 2, yes I would just go to $x+8 = \pm 1$