I know that the fibre of $A_{g,n}$ over $\mathbf{F}_p$ is quasiprojective (of what dimension?). Can one exhibit some smooth projective subvarieties of high dimension in it? What are references for its geometry?

The dimension is $g(g+1)/2$. The supersingular locus gives a large projective subvariety but I don't recall whether it is smooth or not. For references, look up the many papers of F. Oort. 


Over any field there are complete subvarieties of dimension $g1$. Proof: the Satake compactification $A_g^S$ (which exists over $\mathbb Z$, by FaltingsChai) has boundary of codimension $g$. So intersecting $A_g^S$, in some projective embedding, with an appropriate number of hyperplanes will give such subvarieties of $A_g$. 


Let me comment on Felipe Voloch's answer. The supersingular locus (i.e. the locus of $p$rank zero) is indeed complete: it is a closed subvariety of $\overline A_g$ (choose a toroidal compactification) because the $p$rank is lower semicontinuous, and it does not meet the boundary because a torus has positive $p$rank. Moreover it will in fact have the largest possible dimension of a complete subvariety of $A_g$ (in positive characteristic); its dimension is $g(g1)/2$. Indeed if there exists a complete subvariety of dimension $d$ and $\eta$ is an ample divisor class, then $\eta^d \neq 0$. Now $\lambda_1$ (1st Chern class of Hodge bundle) is ample and $\lambda_1^{1+g(g1)/2}$ vanishes. For this see van der Geer, Gerard. Cycles on the moduli space of abelian varieties. Moduli of curves and abelian varieties, 6589, Aspects Math., E33, Vieweg, Braunschweig, 1999. MR1722539 In characteristic zero there is no complete subvariety of $A_g$ of this dimension. Hence the largest dimension of a compact subvariety depends on the characteristic! Keel, Sean; Sadun, Lorenzo. Oort's conjecture for $A_ g\otimes\Bbb C$. J. Amer. Math. Soc. 16 (2003), no. 4, 887900 (electronic). MR1992828 

