A Nonpositive Dot Product Point

The ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has eccentricity $e=\dfrac 12$, left vertex $A$, and lower vertex $B$; let $C$ be the midpoint of the segment $OB$ ($O$ the origin), and suppose the area of $\triangle ABC$ is $\dfrac{3\sqrt 3}2$.

1. Find the equation of the ellipse. 2. A variable line through $C$ meets the ellipse at two points $P$ and $Q$. Determine whether there exists a point $T$ on the $y$-axis such that $\overrightarrow{TP}\cdot\overrightarrow{TQ}\leqslant 0$ always holds; if so, find the range of the $y$-coordinate of $T$, and if not, explain why.