If $R$ is a non-zero ring such that $R \simeq R \oplus R$ as $R$-modules. Then $K(R) \simeq \ast$. The ring map $R \to 0$ induces an equivalence on $K$-theory but $R$ and 0 are not Morita equivalent.

If you take $I$ to be the standard simplicial circle then I think the functor category has infinitely many isomorphism classes. However the indexing simplicial set is not a quasi-category, so this is maybe not what you want.

Quillen's paper "Characteristic classes of representations" (link below) deals with this question from a cohomological perspective. Check out theorem 2'. I hope you have access via the link to Springer: link.springer.com/content/pdf/10.1007/BFb0080002.pdf

If $n=k$ both source and target are free $R$-modules of rank $1$ and you have a surjective $R$-module map between them.

You could for instance take a look at chapter IV of the K-book: math.rutgers.edu/~weibel/Kbook.html

However, I'm not sure that this would capture the topology in a good way if your ring were say the $p$-adic numbers with their usual topology. Maybe someone else can comment on that.

First of all I think you can drop the $\ast$. The algebraic K-groups of a ring A are the homotopy groups of a certain space (or spectrum) K(A) which is defined in terms of the category of finitely generated projective A-modules. If A has a topology you can incorporate this into the construction in a way that (should) give the usual thing for Banach/C*-algebras and definitely gives the right thing for discrete rings.

If you take Segal's $\Gamma$-space construction or Waldhausen's $S.$-construction of $K$-theory and topologize your hom-sets using the topology of your algebra I think you get something that satisfies your criteria, but I don't know if it behaves the way you want on topological $\ast$-algebras that are neither discrete nor $C^{\ast}$. Which kinds of algebras do you have in mind?

A proof of Devissage that only uses the S-construction can be found in Ross Staffeldt's paper "On fundamental theorems of algebraic K-theory". Anyway, Waldhausen gives a natural zig-zag of weak equivalences between the S and the Q-constructions.

The additivity theorem in K-theory says that for an exact category $\mathscr{C}$ the functor $s\times t \colon SES(\mathscr{C}) \to \mathscr{C} \times \mathscr{C}$ induces a homotopy equivalence on K-theory spaces. Here $SES(\mathscr{C})$ is the category of short exact sequences in $\mathscr{C}$ and $s$ and $t$ project on the first and last elements of a sequence, respectively. Is there something like this in L-theory?