The only really "economic" cell structures for $K(\pi,n)$'s that I know is the one with a single cell in each dimension for $K(\mathbb Z/n\mathbb Z,1)$ and the one with a single cell in each even dimension for $K(\mathbb Z,2)$.

Vaguely I remember attending a talk where some lower bounds on numbers of cells in each dimension for Eilenberg-MacLane spaces of cyclic groups were given. Now that I needed this again, I only could find the text "Small CW-models for Eilenberg-MacLane spaces" by Clemens Berger, which contains among other things a cell complex for $K(\mathbb Z/2\mathbb Z,2)$ with 1,0,1,1,2,3,5,8,13,21,... cells (probably the Fibonacci sequence) and a cell complex for $K(\mathbb Z/2\mathbb Z,3)$ with 1,0,0,1,1,2,4,7,13,24,... cells.

What is the current state of the art?

Are there for example any manageable dimensionwise finite cell structures for $K(\mathbb Z,3)$ or $K(\mathbb Z,4)$ or $K(\mathbb Z/n\mathbb Z,2)$ known?

(By manageable I mean... well it is up to you :) )

Is there a geometric construction similar to the real/complex/quaternionic projective spaces known for any other spaces aside of $BO(1)=\mathbb R P^\infty=K(\mathbb Z/2\mathbb Z,1)$, $BU(1)=\mathbb C P^\infty=K(\mathbb Z,2)$ and $B(\textrm{unit quaternions})=\mathbb H P^\infty$? (The latter is of course not any Eilenberg-MacLane space but...)

(Well there are also Grassmanians with their Schubert cells but I mean something as Eilenberg-MacLaneish as possible :) )

Are there any interesting lower bounds on the numbers of cells of each given dimension in a $K(\pi,n)$ known?

And yes of course there is the whole ocean of nonabelian groups with very nice finite-dimensional classifying spaces but I mostly mean $n>1$ and, respectively, abelian groups...

(added after two answers below)

As Jeff Strom and Will Sawin indicate in their answers, homology groups provide the lower bounds, and it is more or less straightforward to arrange for a cell complex with prescribed homology, with minimal possible numbers of cells. Still the question remains (for me) whether such an "absolutely minimal" CW-complex with the correct homotopy type exists.