A Locus from Midpoints of a Rectangle

In the rectangle $ABCD$, $|AB|=4$ and $|BC|=2\sqrt{3}$. Let $E,F,G,H$ be the midpoints of $AB,BC,CD,DA$ respectively, and let $O$ be the center of the rectangle. Set up coordinates with $O$ as the origin, the $x$-axis along line $HF$ and the $y$-axis along line $GE$, so that

Moving points $R$ and $S$ lie on the lines $HF$ and $BC$ respectively and satisfy $\overrightarrow{OR}=\lambda\,\overrightarrow{OF}$ and $\overrightarrow{CS}=\lambda\,\overrightarrow{CF}$ for $\lambda\in\mathbb{R}$. The lines $ER$ and $GS$ intersect at $P$.

1. Prove that $P$ lies on a fixed ellipse, and find the equation of this ellipse.

2. When $\lambda=\dfrac{1}{2}$, a line $l$ through $R$ (not coinciding with the $x$-axis) meets the ellipse from part 1 at two points $M$ and $N$. From $N$ drop a perpendicular to the line $x=4$, with foot $Q$. Let the line $MQ$ meet the $x$-axis at $K$. Find the maximum area of $\triangle KMR$.