A Tangent Slope Product on an Ellipse

The ellipse $C:\dfrac{y^2}{a^2}+\dfrac{x^2}{b^2}=1$ ($a>b>0$) has eccentricity $\dfrac{\sqrt 3}2$ and minor axis of length $2$; let $P$ be its right vertex. Consider the circle $Q:x^2+(y+1)^2=t^2$ ($0<t<1$); from $P$, the two tangent lines to the circle $Q$ meet the ellipse at points $A$ and $B$ respectively (both distinct from $P$).

1. Find the equation of the ellipse $C$. 2. As $t$ varies, is the product of the slopes of $PA$ and $PB$ constant? If so, find it; if not, explain why. 3. For a given $t$, let $d$ be the maximum distance from a point of the ellipse to the line $AB$. As $t$ varies, find the maximum of $d$ and the value of $t$ at which it occurs.