I tried reading about the definition of a pre-additive $\infty$-category, there it is said that a pointed $\infty$-category is pre-additive if all finite products and co-products exist, and the canonical map $X\sqcup Y\longrightarrow X\times Y$ is an equivalence for any pair of objects of the $\infty$-category.
My question is: what are the objects of an $\infty$-category?
The definition I learned says that an $\infty$-category is a simplicial set satisfying a certain property. So I guess my question is not about $\infty$-categories but rather about simplicial sets.
Now, a simplicial set is a functor from the simplex category to sets. Are its objects basically its 1-simplices?
If so, is the isomorphism of products and co-products also a property of a pre-additive's $n$-simplices? Or is it particular specifically for $1$-simplices?
You're probably talking about quasicategories as a model for $\infty$-category theory. Given a quasicategory $C$, the $0$-simplices of $X$ are the objects, and the $1$-simplices are the $1$-morphisms. (Depending on your way of looking at higher morphisms, the $2$-simplices might or might not be $2$-morphisms. Usually we follow the globular picture of a higher morphism, and hence the $2$-simplices are not exactly the $2$-simplices. The same applies to $n$-simplices for $n\geq 2$.)
Therefore $X$ and $Y$ in the definition of a preadditive $\infty$-category are just $0$-simplices of the quasicategory (and the morphism in question is a $1$-simplex).