Abelian categories

Which of these categories are abelian? (Choose all that apply.)

In the proof that a monic epimorphism $f : X \to Y$ in an abelian category is an isomorphism, we write $f = \ker g$ for some $g : Y \to Z$ and conclude $g = 0$. Which fact gives $g = 0$?

A morphism of short exact sequences of finite-dimensional vector spaces has $\dim \ker\alpha = 1$, $\dim\ker\beta = 2$, $\dim\ker\gamma = 3$, $\dim\operatorname{coker}\alpha = 2$ and $\dim\operatorname{coker}\beta = 1$. What is $\dim\operatorname{coker}\gamma$?

In the five lemma (rows $A_1 \to \cdots \to A_5$ and $B_1 \to \cdots \to B_5$, vertical maps $h_1, \dots, h_5$), which hypotheses does the proof that $h_3$ is injective use?

Which of the following are legitimate uses of the Freyd–Mitchell embedding theorem? (Choose all that apply.)

On $X = \mathbb{C} \setminus \{0\}$ consider $0 \to \underline{\mathbb{Z}} \to \mathcal{O} \xrightarrow{\exp} \mathcal{O}^\times \to 0$. Which statement is true?

The Lean theorem short_five_injective concludes Function.Injective β. Which set of hypotheses (besides commutativity) does it use?

For an integer matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, viewed as a map $\mathbb{Z}^2 \to \mathbb{Z}^2$, the cokernel in $\mathbf{Ab}$ is $\mathbb{Z}/d_1 \oplus \mathbb{Z}/d_2$ (with $\mathbb{Z}/0 = \mathbb{Z}$), where $d_1 = \gcd(a, b, c, d)$ and $d_1 d_2 = |\det M|$ (if $M = 0$, the cokernel is $\mathbb{Z}^2$). Write cokernel_2x2(a, b, c, d) returning (torsion, free_rank): the sorted list of torsion orders greater than $1$, and the rank of the free part.