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...