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, $R=F_{p}\left[\left[G\right]\right]$.

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.