Does earth's Umbra reach Sun-Earth L2? Moon can be fully eclipsed by the virtue of being fairly close to Earth. Any body distant enough will not be eclipsed fully, Earth's disc not fully covering Sun's disc. How's that for anything at the L2 point? Will it be lit by Sun's corona or will it bask in deep shadow of Earth?

Bonus question: if Umbra does reach there, how large a body could remain fully shaded at L2? What's the Umbra's radius there?
The Lagrangian point $L_2$ is very close to the most distant point from Earth with an umbra.
$L_2$ is like the radius of the Hill sphere at $r=a\sqrt[3]{\frac{m}{3M}}$ for circular orbits, with $m$ the mass of Earth, $M$ the mass of the Sun, and $a$ the distance Earth-Sun. The ratio $\frac{m}{3M}$ of the Earth and the triple mass of the Sun is almost exactly $10^{-6}$, the cubic root hence $0.01$.

The diameter ratio of Earth and Sun is about $1/109$. Therefore the umbra of Earth ends near $92\%$ the distance to $L_2$.

The answer to another bonus question would then be: If Earth would be $9\%$ larger in diameter, but with the same mass, its umbra would end almost exactly at $L_2$.

Earth's orbit isn't perfectly circular, but the aphel/perihel ratio of about $1.04$ is insufficient to question the result qualitatively.
The error of the implicite assumptions $\tan x=x=\sin x$ is negligible at the considered level of accuracy.