This is not an answer, but I'd like to point out that the concepts in question are projective, although they have a special Euclidean case.
Consider the diagram below. Start with a conic $\gamma$(green), a triangle $ABC$ inscribed in $\gamma$, and a line $\omega$ (black dot-dashed). Let $X$ be the polar of $\omega$ wrt the conic, and draw a line (dotted) through $X$ that meets $\omega$ at $P,P'$. Let the dashed lines through $P'$ meet the respective lines from $P$ to $A,B,C$ at $\omega$. Then the dashed lines meet the triangle sides at collinear points (red), and $X$ lies on this line.
The OP is the special case when $\omega$ is the projective line at infinity.
So, if the line in OP Question 1 is known, and anybody is trying to hunt it down, it may be in the projective geometry literature.
