= Solution
Let $R$ be the total number of rejected <null hypotheses>, and let $V$ count rejected null hypotheses that are in fact true. The <false discovery proportion> and <false discovery rate> are
$$
\boxed{\operatorname{FDP}=\frac{V}{R\vee1},\qquad \operatorname{FDR}=\mathbb E\left[\frac{V}{R\vee1}\right].}
$$
Thus the <false discovery proportion> is zero when there are no rejections, and the <false discovery rate> averages the realized proportion of erroneous rejections.
For the <Benjamini-Hochberg procedure>, order the <p-values> as $P_{(1)}\le\cdots\le P_{(m)}$ and define
$$
R=\max\bigl(\{k\in\{1,\ldots,m\}:P_{(k)}\le\alpha k/m\}\cup\{0\}\bigr).
$$
Reject every hypothesis whose <p-value> satisfies $P_i\le\alpha R/m$ when $R>0$, and reject none otherwise. There are exactly $R$ such hypotheses: if a further <p-value> also met the threshold, its larger rank would satisfy its own, still larger, threshold, contradicting maximality. Under the stated <independence> and uniform true-null <p-values>, this procedure has <false discovery rate> $\alpha m_0/m\le\alpha$.
For the first modified <Benjamini-Hochberg procedure>, remove $P_1$, order the remaining $m-1$ <p-values> as $Q_{(1)},\ldots,Q_{(m-1)}$, and use the shifted critical values $\alpha(j+1)/m$. Define its rejection count by
$$
R_{-1}=\max\bigl(\{j\in\{1,\ldots,m-1\}:Q_{(j)}\le\alpha(j+1)/m\}\cup\{0\}\bigr).
$$
Equivalently, insert a zero in place of $P_1$ and run the ordinary <Benjamini-Hochberg procedure> on $m$ values; its total rejection count is $R^{(1)}=R_{-1}+1$. For $r=1,\ldots,m$, the <Benjamini-Hochberg leave-one-out identity> in event form is
$$
\boxed{\{P_1\le\alpha r/m,\ R=r\}=\{P_1\le\alpha r/m,\ R_{-1}=r-1\}.}
$$
To prove it, suppose first that the left event holds. Then $P_1$ is among the rejected values. Lowering it to zero leaves all ordered <p-values> at ranks greater than $r$ unchanged, so no new rank can meet its threshold. Rank $r$ still meets its threshold, giving $R^{(1)}=r$. Conversely, if $R^{(1)}=r$ and $P_1\le\alpha r/m$, the $r-1$ rejected remaining values and $P_1$ all meet the original rank-$r$ threshold. Hence the original count is at least $r$. Lowering a <p-value> cannot decrease the rejection count, so the original count is also at most $R^{(1)}=r$. This proves equality of the events, including possible ties.
For the second modified <Benjamini-Hochberg procedure>, remove both $P_1,P_2$, order the remaining <p-values> as $T_{(j)}$, and use shifted critical values $\alpha(j+2)/m$. Its count is
$$
R_{-12}=\max\bigl(\{j\in\{1,\ldots,m-2\}:T_{(j)}\le\alpha(j+2)/m\}\cup\{0\}\bigr).
$$
Replacing $P_1,P_2$ by zeros in the full procedure gives a total of $R^{(12)}=R_{-12}+2$ rejections. The same argument gives the <Benjamini-Hochberg leave-two-out identity>, for $r=2,\ldots,m$:
$$
\boxed{\{P_1\le\alpha r/m,\ P_2\le\alpha r/m,\ R=r\}
=\{P_1\le\alpha r/m,\ P_2\le\alpha r/m,\ R_{-12}=r-2\}.}
$$
In the forward direction both replaced values already lie among the first $r$, so ordered ranks greater than $r$ remain unchanged. In the reverse direction reinserting the two values below the rank-$r$ threshold still leaves at least $r$ qualifying values, while monotonicity under lowering bounds the original count above by $r$. For $r=1$, the event of two <p-values> meeting the displayed threshold while only one is rejected is empty. For $m=1$ only the first modification is needed.
To calculate the second moment, write $I_i=\mathbf1\{i\text{ rejected}\}$ for a true null. Expanding the square gives
$$
V^2=\sum_{i=1}^{m_0}I_i+\sum_{\substack{1\le i,j\le m_0\\i\ne j}}I_iI_j.
$$
For a true-null index $i$, let $R_{-i}$ denote the first modified count after removing $P_i$. It is a function only of the remaining <p-values>, and so is independent of the uniform variable $P_i$. The first event identity therefore yields
$$
\mathbb E\frac{I_i}{(R\vee1)^2}
=\sum_{r=1}^m\frac1{r^2}\mathbb P(P_i\le\alpha r/m,\ R_{-i}=r-1)
=\frac\alpha m\sum_{r=1}^m\frac{\mathbb P(R_{-i}=r-1)}r
=\frac\alpha m\mathbb E\frac1{R_{-i}+1}.
$$
For distinct true-null indices $i,j$, the two independent uniform <p-values> are independent of the second modified count $R_{-ij}$. The second event identity gives
$$
\mathbb E\frac{I_iI_j}{(R\vee1)^2}
=\sum_{r=2}^m\frac1{r^2}\left(\frac{\alpha r}{m}\right)^2\mathbb P(R_{-ij}=r-2)
=\frac{\alpha^2}{m^2}.
$$
The last sum is one because $R_{-ij}$ always takes a value in $\{0,\ldots,m-2\}$. The ordered pairs in the expansion of $V^2$ number $m_0(m_0-1)$, with no factor of one half.
The true-null <p-values> are identically distributed and independent, so permutations among their indices leave the law of the full collection unchanged. Thus every $R_{-i}$ with $i\le m_0$ has the same distribution as $R_{-1}$, even though the false-null <p-values> need not be identically distributed. Combining the diagonal and off-diagonal contributions proves the <second moment of the Benjamini-Hochberg false discovery proportion>:
$$
\boxed{\mathbb E(\operatorname{FDP}^2)=\frac{\alpha m_0}{m}\mathbb E(A)+\frac{\alpha^2m_0(m_0-1)}{m^2},\qquad A=\frac1{R_{-1}+1}.}
$$
If the first modified count is instead defined to include the inserted zero, the same answer is $A=1/R^{(1)}$. When $m_0=0$, both sides are zero; when $m_0=1$, there are no off-diagonal terms. Finally, repeating the first calculation with denominator $R\vee1$ rather than its square gives $\mathbb E[I_i/(R\vee1)]=\alpha/m$, which verifies the claimed exact <false discovery rate>.
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