Proof.
If $\mathfrak m = (x, y)$, then $X$ is covered by the spectra of the affine blowup algebras $A[\frac{\mathfrak m}{x}]$ and $A[\frac{\mathfrak m}{y}]$ because $x$ and $y$ placed in degree $1$ generate the Rees algebra $\bigoplus \mathfrak m^ n$ over $A$. See Divisors, Lemma 31.32.2 and Constructions, Lemma 27.8.9. Since $X$ is separated by Constructions, Lemma 27.8.8 we see that cohomology of quasi-coherent sheaves vanishes in degrees $\geq 2$ by Cohomology of Schemes, Lemma 30.4.2.
Let $i : E \to X$ be the exceptional divisor, see Divisors, Definition 31.32.1. Recall that $\mathcal{O}_ X(-E) = \mathcal{O}_ X(1)$ is $f$-relatively ample, see Divisors, Lemma 31.32.4. Hence we know that $H^1(X, \mathcal{O}_ X(-nE)) = 0$ for some $n > 0$, see Cohomology of Schemes, Lemma 30.16.2. Consider the filtration
\[ \mathcal{O}_ X(-nE) \subset \mathcal{O}_ X(-(n - 1)E) \subset \ldots \subset \mathcal{O}_ X(-E) \subset \mathcal{O}_ X \subset \mathcal{O}_ X(E) \]
The successive quotients are the sheaves
\[ \mathcal{O}_ X(-t E)/\mathcal{O}_ X(-(t + 1)E) = \mathcal{O}_ X(t)/\mathcal{I}(t) = i_*\mathcal{O}_ E(t) \]
where $\mathcal{I} = \mathcal{O}_ X(-E)$ is the ideal sheaf of $E$. By Lemma 54.3.1 we have $E = \mathbf{P}^1_\kappa $ and $\mathcal{O}_ E(1)$ indeed corresponds to the usual Serre twist of the structure sheaf on $\mathbf{P}^1$. Hence the cohomology of $\mathcal{O}_ E(t)$ vanishes in degree $1$ for $t \geq -1$, see Cohomology of Schemes, Lemma 30.8.1. Since this is equal to $H^1(X, i_*\mathcal{O}_ E(t))$ (by Cohomology of Schemes, Lemma 30.2.4) we find that $H^1(X, \mathcal{O}_ X(-(t + 1)E)) \to H^1(X, \mathcal{O}_ X(-tE))$ is surjective for $t \geq -1$. Hence
\[ 0 = H^1(X, \mathcal{O}_ X(-nE)) \longrightarrow H^1(X, \mathcal{O}_ X(-tE)) = H^1(X, \mathcal{O}_ X(t)) \]
is surjective for $t \geq -1$ which proves (2).
Let $\mathcal{F}$ be globally generated. This means there exists a short exact sequence
\[ 0 \to \mathcal{G} \to \bigoplus \nolimits _{i \in I} \mathcal{O}_ X \to \mathcal{F} \to 0 \]
Note that $H^1(X, \bigoplus _{i \in I} \mathcal{O}_ X) = \bigoplus _{i \in I} H^1(X, \mathcal{O}_ X)$ by Cohomology, Lemma 20.19.1. By part (2) we have $H^1(X, \mathcal{O}_ X) = 0$. If $\mathcal{F}(1)$ is globally generated, then we can find a surjection $\bigoplus _{i \in I} \mathcal{O}_ X(-1) \to \mathcal{F}$ and argue in a similar fashion. In other words, part (3) follows from part (2).
For part (4) we note that for all $n$ large enough we have $\Gamma (X, \mathcal{O}_ X(n)) = \mathfrak m^ n$, see Cohomology of Schemes, Lemma 30.14.3. If $n \geq 0$, then we can use the short exact sequence
\[ 0 \to \mathcal{O}_ X(n) \to \mathcal{O}_ X(n - 1) \to i_*\mathcal{O}_ E(n - 1) \to 0 \]
and the vanishing of $H^1$ for the sheaf on the left to get a commutative diagram
\[ \xymatrix{ 0 \ar[r] & \mathfrak m^{\max (0, n)} \ar[r] \ar[d] & \mathfrak m^{\max (0, n - 1)} \ar[r] \ar[d] & \mathfrak m^{\max (0, n)}/\mathfrak m^{\max (0, n - 1)} \ar[r] \ar[d] & 0\\ 0 \ar[r] & \Gamma (X, \mathcal{O}_ X(n)) \ar[r] & \Gamma (X, \mathcal{O}_ X(n - 1)) \ar[r] & \Gamma (E, \mathcal{O}_ E(n - 1)) \ar[r] & 0 } \]
with exact rows. In fact, the rows are exact also for $n < 0$ because in this case the groups on the right are zero. In the proof of Lemma 54.3.1 we have seen that the right vertical arrow is an isomorphism (details omitted). Hence if the left vertical arrow is an isomorphism, so is the middle one. In this way we see that (4) holds by descending induction on $n$.
Finally, we prove (5) by descending induction on $n$ and the sequences
\[ 0 \to \mathcal{O}_ X(n) \to \mathcal{O}_ X(n - 1) \to i_*\mathcal{O}_ E(n - 1) \to 0 \]
Namely, for $n \geq -1$ we already know $H^1(X, \mathcal{O}_ X(n)) = 0$. Since
\[ H^1(X, i_*\mathcal{O}_ E(-2)) = H^1(E, \mathcal{O}_ E(-2)) = H^1(\mathbf{P}^1_\kappa , \mathcal{O}(-2)) \cong \kappa \]
by Cohomology of Schemes, Lemma 30.8.1 which has length $1$ as an $A$-module, we conclude from the long exact cohomology sequence that (5) holds for $n = -2$. And so on and so forth.
$\square$
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