Thank you! I would like to see some kind of a universal way to pick out "small" objects (=finitely presented or finitely built or things like that) from a category.

Thank you! My only problem is the following. Point in a general category should be either the initial object or the terminal object. In the category of $R$-mod or in the category of (co)chain complexes, however, it is the trivial object (0), and any colimit of that is the trivial object, so I won't get any interesting objects.

This might help. The first step in defining the Grothendieck group of a scheme is to restrict the objects to small ones (i.e. to coherent sheaves), otherwise the group will be trivial. Now, I would like to define the Grothendieck group of some categories. So the first step would be to define small objects in it. For example, for the category of k-schemes, I would like to obtain Dream 1.

@Muro The only notion I know is the notion of "compact object", but the examples I know have nothing to do with actual schemes or spaces (see Yuan's comment). I would like to find some similar notion which gives me Dream 1 and Dream 2. I do not know how to state the question more clearly.

@Lin Can I ask you why? If I think of my model category as an $(\infty, 1)$-category, then for me it seems more natural to take the hom spaces rather the hom set.

@Yuan, thank you! That's a good observation, I change the question.

@Muro I do not see why the model structure is irrelevant. The hom spaces do not change while changing the model structure?

I do not think one should expect a formula involving the $H^k(\wedge ^j)$ and $H^l(\mathcal{End}(E))$.

They are isomorphic as algebra objects in the derived category of $X$. This phenomenon is not unique for the diagonal embedding. If you have a closed embedding $i:X\rightarrow Y$, and $\mathcal{F}$ is any vector bundle, then $i^*i_*\mathcal{F}$ and $F\otimes Sym(N^{\vee}[1])$ are isomorphic, if $X$ splits in $Y$ (so the normal sequence splits). (see When is the self-intersection of a subvariety a fubration? by Arinkin and Caldararu)