An Equal-Sums Locus

The ellipse $G \colon \dfrac{x^2}{6} + \dfrac{y^2}{b^2} = 1$ (with $0 < b < \sqrt{6}$) has foci $F_1, F_2$ and minor-axis endpoints $B_1, B_2$. A point $P$ on $G$ satisfies

Determine, with proof, which of the following are true as $b$ varies:

1. the locus of such points $P$ is symmetric about the $y$-axis; 2. for some $b$, exactly two points $P$ of the ellipse satisfy the condition; 3. the minimum value of $|OP|$ is $2$.