I think I have an example.
Fix a chain of fields $k_\alpha$ indexed by ordinals $\alpha$, where $k_\alpha\subset k_\beta$ is an infinite field extension for all pairs $\alpha<\beta$ of ordinals.
First I'll define an "abelian category" which has large Hom-sets.
An object $V$ will consist of a $k_\alpha$-vector space $V(\alpha)$ for each ordinal $\alpha$, together with a $k_\alpha$-linear map $v_{\alpha,\beta}:V(\alpha)\to V(\beta)$ for each pair $\alpha<\beta$ of ordinals, such that $v_{\beta,\gamma}\circ v_{\alpha,\beta}=v_{\alpha,\gamma}$ whenever $\alpha<\beta<\gamma$.
A morphism $\theta:V\to W$ will consist of a $k_\alpha$-linear map $\theta_\alpha:V(\alpha)\to W(\alpha)$ for each $\alpha$, such that $w_{\alpha,\beta}\circ\theta_\alpha=\theta_\beta\circ v_{\alpha,\beta}$ for all $\alpha<\beta$.
Now let's say that an object $V$ is "$\alpha$-good" if, for every $\beta>\alpha$, $V(\beta)$ is generated as a $k_\beta$-vector space by the image of $v_{\alpha,\beta}$, and that $V$ is "good" if it is $\alpha$-good for some $\alpha$. If $V$ is $\alpha$-good, then any morphism $\theta:V\to W$ is determined by $\theta_\gamma$ for $\gamma\leq\alpha$, so the full subcategory $\mathfrak{C}$ of good objects has small Hom-sets.
It's straightforward to check that $\mathfrak{C}$ is an abelian category, and it has small coproducts in the obvious way, where $\left(\coprod_{i\in I} V_i\right)(\alpha)=\coprod_{i\in I} V_i(\alpha)$.
I claim that $\mathfrak{C}$ does not have all small products.
For any $\alpha$, let $P_{\alpha}$ be the ($\alpha$-good) object with
$$P_\alpha(\beta)=\begin{cases}0&\mbox{if }\beta<\alpha\\k_\beta&\mbox{if }\alpha\leq\beta\end{cases}$$
and obvious inclusion maps. Then for any object $W$, $\operatorname{Hom}_{\mathfrak{C}}(P_\alpha,W)$ is naturally isomorphic to $W(\alpha)$ (i.e., $P_\alpha$ represents the functor $W\mapsto W(\alpha)$ from $\mathfrak{C}$ to $k_\alpha$-vector spaces), and if $\alpha<\beta$ then the map
$$W(\alpha)=\operatorname{Hom}_{\mathfrak{C}}(P_\alpha,W)\to\operatorname{Hom}_{\mathfrak{C}}(P_\beta,W)=W(\beta),$$
induced by the obvious inclusion $P_\beta\to P_\alpha$, is just $w_{\alpha,\beta}$.
Suppose $W$ were the product in $\mathfrak{C}$ of of a countable number of copies of $P_0$. Since it's an object of $\mathfrak{C}$, $W$ must be $\alpha$-good for some $\alpha$.
Then
$$W(\alpha)=\operatorname{Hom}_{\mathfrak{C}}(P_\alpha,W)=\prod_{i\in\mathbb{N}}k_\alpha$$
and for $\beta>\alpha$
$$W(\beta)=\operatorname{Hom}_{\mathfrak{C}}(P_\beta,W)=\prod_{i\in\mathbb{N}}k_\beta.$$
But then $W(\beta)$ is not generated as a $k_\beta$-vector space by the image of the natural map $w_{\alpha,\beta}:W(\alpha)\to W(\beta)$, since $k_\beta$ is an infinite extension of $k_\alpha$, contradicting the $\alpha$-goodness of $W$.
The following pair of examples follows the idea of Jeremy Rickard suggested in a comment on Math Stack Exchange under the link. Inverting the arrows, it suffices to construct an example of abelian category $\mathcal A$ such that the category of right exact functors $\mathcal A \to \mathcal Ab$ or $\mathcal A \to k{-}\mathcal{V}ect$ is not abelian (where $k$ is a field).
Let $p$ be a prime number, and let $\mathcal A_p$ be the abelian category of finite abelian $p$-groups. Let me start with a very brief introductory discussion of the category of left exact functors $\mathcal A_p\to\mathcal Ab$. The category of finite abelian $p$-groups is self-dual (by Pontryagin duality, taking a finite abelian $p$-group $A$ to the group $\operatorname{Hom}_{\mathbb Z}(A,\mathbb Q/\mathbb Z)$). So the category of left exact functors $\mathcal A_p\to\mathcal Ab$ is equivalent to the category of left exact functors $\mathcal A_p^{op}\to\mathcal Ab$.
The category of left exact functors $E\colon\mathcal A_p^{op}\to\mathcal Ab$ is equivalent to the category of $p$-primary torsion abelian groups $T$. To a $p$-primary torsion abelian group $T$ one assigns the contravariant functor $E_T$ taking a finite abelian $p$-group $A$ to the abelian group $\operatorname{Hom}_{\mathbb Z}(A,T)$. To construct the inverse functor, consider the projective system of finite abelian $p$-groups $\mathbb Z/p\mathbb Z \leftarrow \mathbb Z/p^2\mathbb Z\leftarrow \mathbb Z/p^3\mathbb Z\leftarrow\dotsb$ and apply the contravariant functor $E$ to it. The $p$-primary torsion abelian group $T_E$ corresponding to $E$ is the inductive limit of the sequence $E(\mathbb Z/p\mathbb Z)\to E(\mathbb Z/p^2\mathbb Z)\to E(\mathbb Z/p^3\mathbb Z)\to\dotsb$. From the left exactness property of the functor $E$ one can see that the group $E_m=E(\mathbb Z/p^m\mathbb Z$) is identified with the subgroup of elements annihilated by $p^m$ in the group $E_n$, for every $n\ge m$.
The category of $p$-primary torsion abelian groups is abelian. Hence so is the category of left exact functors $\mathcal A_p\to \mathcal Ab$ (as we know it should be by the additive sheaf theory).
Now let us turn to right exact functors $\mathcal A_p\to \mathcal Ab$, which we are really interested in. Let $F\colon\mathcal A_p\to\mathcal Ab$ be a right exact functor. Then $F_n=F(\mathbb Z/p^n\mathbb Z)$ is an abelian group annihilated by the multiplication with $p^n$. For any $n\ge m$, from the right exact sequence $\mathbb Z/p^n\mathbb Z\to \mathbb Z/p^n\mathbb Z \to \mathbb Z/p^m\mathbb Z\to 0$ we get an isomorphism $F_n/p^mF_n\cong F_m$. The functor $F$ is uniquely determined by the sequence of abelian groups $F_n$ together with these isomorphisms. Indeed, for every abelian group $A$ annihilated by $p^n$ one has $F(A)=F_n\otimes_{\mathbb Z/p^n\mathbb Z} A$.
To any right exact functor $F\colon\mathcal A_p\to\mathcal Ab$, one assigns the abelian group $C_F=\varprojlim_n F_n$. This assignment is an equivalence between the category of right exact functors $\mathcal A_p\to\mathcal Ab$ and the full subcategory in abelian groups consisting of all the $p$-separated $p$-complete abelian groups, i.e., abelian groups $C$ for which the natural map $C\to\varprojlim_n C/p^nC$ is an isomorphism. The inverse functor assigns to an abelian group $C$ the functor $F_C$ taking a finite abelian $p$-group $A$ to the abelian group $C\otimes_{\mathbb Z}A$. The corresponding groups $F_n$ are $F_n=C/p^nC$.
The category of all $p$-separated $p$-complete abelian groups is not abelian. There is a closely related (locally $\aleph_1$-presentable) abelian category of "weakly $p$-complete" or "Ext-$p$-complete" abelian groups, but it is a different (bigger) category: all $p$-separated $p$-complete abelian groups are Ext-$p$-complete, and all Ext-$p$-complete abelian groups are $p$-complete, but they need not be $p$-separated. In other words, the map $C\to \varprojlim_n C/p^nC$ for an Ext-$p$-complete abelian group $C$ is always surjective, but it need not be injective.
Similarly, let $k$ be a field, and let $\mathcal A_k$ be the abelian category of finite-dimensional $k$-vector spaces $V$ endowed with a nilpotent $k$-linear operator $x\colon V\to V$. Let $G\colon\mathcal A_k\to k{-}\mathcal{V}ect$ be a $k$-linear right exact functor. Then $G_n=G(k[x]/x^nk[x])$ is a $k[x]/x^nk[x]$-module. Just as in the first example above, one constructs a natural isomorphism $G_n/x^mG_n\cong G_m$ for every $n\ge m$. The functor $G$ is uniquely determined by the sequence of modules $G_n$ together with these isomorphisms.
To any $k$-linear right exact functor $G\colon\mathcal A_k\to k{-}\mathcal{V}ect$, one assigns the $k[x]$-module (or, if one wishes, $k[[x]]$-module) $D_G=\varprojlim_n G_n$. This assignment is an equivalence between the category of $k$-linear right exact functors $G\colon\mathcal A_k\to k{-}\mathcal{V}ect$ and the full subcategory in $k[x]$-modules (or, equivalently, in $k[[x]]$-modules) consisting of all the $x$-separated and $x$-complete modules, i.e., $k[x]$-modules or $k[[x]]$-modules $D$ such that the natural map $D\to\varprojlim_n D/x^nD$ is an isomorphism. The inverse functor assigns to a $k[x]$-module $D$ the functor $G_D$ taking a $k$-finite-dimensional $k[x]$-module $B$ with $x$ acting by a nilpotent operator to the $k$-vector space $D\otimes_{k[x]}B$. The corresponding modules $G_n$ are $G_n=D/x^nD$.
Taking $k=\mathbb Z/p\mathbb Z$ or $k=\mathbb Q$, one can drop the adjective "$k$-linear" before the words "right exact functor" (as any abelian group homomorphism between $k$-vector spaces is a $k$-linear map, so any additive functor between $k$-linear categories is $k$-linear, for such fields $k$).
As in the first example, the category of all $x$-separated $x$-complete $k[x]$-modules is not abelian. There is a closely related (locally $\aleph_1$-presentable) abelian category of "weakly $x$-complete" or "Ext-$x$-complete" $k[x]$-modules, but it is a different (bigger) full subcategory in $k[x]{-}\mathcal{M}od$. Any Ext-$x$-complete $k[x]$-module is $x$-complete, but it need not be $x$-separated.
The categories of $p$-separated $p$-complete abelian groups and $x$-separated $x$-complete $k[x]$-modules are complete and cocomplete. In fact, they are locally $\aleph_1$-presentable, being reflective and closed under $\aleph_1$-filtered colimits as full subcategories in the locally $\aleph_1$-presentable abelian categories of Ext-$p$-complete abelian groups and Ext-$x$-complete $k[x]$-modules (respectively). So, in particular, all morphisms in these two categories (of right exact functors) have kernels and cokernels. Still, these two categories are not abelian.
The counterexample showing that the categories of $p$-separated $p$-complete abelian groups and $x$-separated $x$-complete $k[x]$-modules are not abelian is now well-known. It has been rediscovered and discussed by many authors, including Example 2.5 in A.-M. Simon, "Approximations of complete modules by complete big Cohen-Macaulay modules over a Cohen-Macaulay local ring", Algebras and Represent. Theory 12, 2009, and Example 3.20 in A. Yekutieli, "On flatness and completion for infinitely generated modules over noetherian rings", Communic. in Algebra 39, 2010, https://arxiv.org/abs/0902.4378 .
The specific assertion that these two categories (or, rather, the first one, but there is no difference), viewed as abstract categories, are not abelian, can be found in Example 2.7(1) in my paper L. Positselski, "Contraadjusted modules, contramodules, and reduced cotorsion modules", Moscow Math. Journ. 17 #3, 2017, https://arxiv.org/abs/1605.03934 .
The above descriptions of two right exact functor categories as categories of separated and complete modules (= "separated contramodules") are the simplest examples to a much more general theory developed in Section 6 of our paper L. Positselski, J. Rosicky, "Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories", Journ. Algebra 483, 2017, https://arxiv.org/abs/1512.08119 . Specifically, the discussion in Remark 6.5 in this paper is relevant.
Best Answer
Here's a simple counterexample for (2).
Let $\mathcal{A}$ be the category of sequences of linear maps $U\stackrel{\alpha}{\to}V\stackrel{\beta}{\to}W$ between vector spaces over a field $k$ such that $\beta\alpha=0$, and let $\mathcal{C}$ be the full subcategory of objects of the form $0\to0\to W$, so that $\mathcal{A}/\mathcal{C}$ is equivalent to the category of linear maps $U\to V$, with the quotient functor sending $U\to V\to W$ to $U\to V$ and the section functor sending $U\to V$ to $U\to V\to0$.
Then $0\to k=k$ is injective in $\mathcal{A}$, but $T(0\to k=k)$ is $0\to k$, which is not injective in $\mathcal{A}/\mathcal{C}$.
For the projective case, you can use the same $\mathcal{A}$, but take $\mathcal{C}$ to be the category of objects $U\to0\to0$, so that $\mathcal{A}/\mathcal{C}$ is again equivalent to the category of linear maps, the quotient functor sends $U\to V\to W$ to $V\to W$, and the section functor sends $V\stackrel{\beta}{\to}W$ to $\ker(\beta)\to V\stackrel{\beta}{\to}W$.
Then $k=k\to0$ is projective in $\mathcal{A}$ but $T(k=k\to0$) is $k\to 0$, which is not projective in $\mathcal{A}/\mathcal{C}$.