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Deformation of (locally) ringed spaces and of their abelian categories of modules

The answer to your second question is "no", I think. Let's assume that sufficiently nice means that it is a smooth algebraic variety over a field of characteristic $0$. Then, as written in ...
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Deformation of (locally) ringed spaces and of their abelian categories of modules

As Jon Pridham notes in the comments, the quote should be understood noncommutatively. In fact, in the introduction Lowen and Van den Bergh write Deformation theory of abelian categories is important ...
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Does the category of commutative and cocommutative Hopf algebras have enough injectives?

Over a field $k$, the answer is yes for injectives; I'm not sure about projectives. Over $\mathbb Z$ or other commutative rings, I really don't know -- the use of the fundamental theorem of coalgebra ...
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Does left-exactness imply semi-additivity?

So I don't know if assuming biproducts exists is enough or not but preserving kernel alone is not enough. Here is a counter-exemple. Let $C$ be the pre-additive category with only one object $*$ and ...
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