(1) Yes, the trivial (final) category, with only one object and one morphism (the identity in the unique object).

(2) If you only asked for colimits of cardinality $<\kappa$ then you would find many examples, e.g. finite abelian groups ($\kappa = \aleph_0$ here). If you insists in $\leq\kappa$ I think you face the same problem as above. Just think of vector spaces of dimension $<\kappa$. This category, up to isomorphism, has $\leq \kappa$ objects but it doesn't have colimits of sice $\leq \kappa$ since the coproduct of $\kappa$ copies of the ground field has dimension exactly $\kappa$.

I've been speaking about colimits in (2) instead of limits, which is what you ask for, so take opposite categories.