When is an OLS parameter unchanged on a subsample?
I think I figured out 1). I am still interested in 2).

OLS sets $b_0$ and $b_1$ in such a way that

$$
(b_0 \ \  b_1) = \arg\min_{a_0, a_1} \sum_j \left(Y_j - a_0 - a_1 X_j\right)^2. 
$$
The first order conditions are
$$
(-2)\sum_j \left(Y_j - b_0 - b_1 X_j\right) = 0 
$$
and
$$
(-2)\sum_j X_j \left(Y_j - b_0 - b_1 X_j\right) = 0.
$$
If observation $i$ is on the regression line then
$$
\begin{align*}
\left(Y_i - b_0 - b_1 X_i\right) &amp; = 0 \\
\\
X_i\left(Y_i - b_0 - b_1 X_i\right) &amp; = 0.
\end{align*}
$$
It follows from this and the original first order conditions that
$$
\begin{align*}
(-2)\sum_{j\neq i} \left(Y_j - b_0 - b_1 X_j\right) &amp; = 0 \\
\\
(-2)\sum_{j\neq i} X_j\left(Y_j - b_0 - b_1 X_j\right) &amp; = 0,
\end{align*}
$$
which are the first order conditions of the subsample's OLS problem.