I don't know if this counts as a "naturally-occuring" algebraic category but as I scanned through my internal list of categories of modules over a ring, I spotted that the following has (1) and (2).

Take G to be the "p-adic dihedral group" that is a semidirect product of the p-adic integers and a cyclic group of order 2 where the latter acts by -1 on the former. Then form the completed group algebra (Iwasawa algebra) with coefficients in the field with p elements, R=F_p[[G]].

Now there are precisely two isomorphism classes of indecomposable projective modules [P_ 1] and [P_ 2] and P_ 1 and P_ 2 are bi-embeddable via monic maps (see section 9.6 of http://www.dpmms.cam.ac.uk/~sjw47/char.pdf for the proof of both these claims). 

It follows that if we take the category of all finitely generated projective modules over R then it consists of objects that are direct sums of m copies of P_ 1 and n copies of P_ 2, say. Two such objects will be bi-embeddable if and only if they have the same value of m+n thus whilst there are arbitrarily large finite collections of pairwise bi-embeddable but pairwise nonisomorphic objects in this category there are no infinite ones.
 
I am sure there will be many more examples along these lines, I just happen to spend a lot of time thinking about Iwasawa algebras.