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 \ref{divisors-lemma-blowing-up-affine} and Constructions, Lemma \ref{constructions-lemma-proj-quasi-compact}. Since $X$ is separated by Constructions, Lemma \ref{constructions-lemma-proj-separated} we see that cohomology of quasi-coherent sheaves vanishes in degrees $\geq 2$ by Cohomology of Schemes, Lemma \ref{coherent-lemma-vanishing-nr-affines}. \medskip\noindent Let $i : E \to X$ be the exceptional divisor, see Divisors, Definition \ref{divisors-definition-blow-up}. Recall that $\mathcal{O}_X(-E) = \mathcal{O}_X(1)$ is $f$-relatively ample, see Divisors, Lemma \ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}. Hence we know that $H^1(X, \mathcal{O}_X(-nE)) = 0$ for some $n > 0$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-kill-by-twisting}. 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 \ref{lemma-blowup} 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 \ref{coherent-lemma-cohomology-projective-space-over-ring}. Since this is equal to $H^1(X, i_*\mathcal{O}_E(t))$ (by Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-cohomology}) 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). \medskip\noindent 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 \ref{cohomology-lemma-quasi-separated-cohomology-colimit}. 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). \medskip\noindent 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 \ref{coherent-lemma-recover-tail-graded-module}. 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 \ref{lemma-blowup} 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 $n$. \medskip\noindent 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 \ref{coherent-lemma-cohomology-projective-space-over-ring} 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.