For the generalization in the first direction, the $p$-rank of the incidence matrix $N$ of an $S(2,k,v)$ is lower bounded by the dimension of the Steinberg module:

$$\operatorname{rank}_2(N)(\operatorname{rank}_2(N)-1) \geq \frac{(v-1)(v-k)}{k} \quad \text{if} \frac{v-k}{k-1} \text{is even}$$
and if further $k$ is odd,
$$\operatorname{rank}_2(N) \geq 1+\sqrt{\frac{(v-1)(v-k)}{k}}.$$
If $\frac{v-k}{k-1}$ is odd, a simple argument shows that
$$\operatorname{rank}_2(N) \geq v-1$$
with equality if and only if $k$ is even.

A more general bound can be proved by applying the result given in the following paper and other known facts:

G. Hillebrandt, The $p$-rank of $(0,1)$-matrices, *J. Combin. Theory, Ser A,* **60** (1992), 131-139.

This and some other facts about the rank of an $S_{\lambda}(2,k,v)$ are explained in the language of design theory in Section 2.4 of *Designs and Their Codes* by E. F. Assmus Jr. and J. D. Key.

As in the case of $S(2,3,v)$s, the lowest rank has been studied in relation to geometric designs. The rank of the incidence matrix of the $2$-design formed by the points and subspaces of projective/affine geometry can be computed by well-known Hamada's formula for many cases, although it is a little cumbersome. A more general result in this direction is available here:

D. B. Chandler, P. Sin, Q. Xiang, The invariant factors of the incidence matrices of points and subspaces in $ \operatorname{PG}(n,q)$ and $ \operatorname{AG}(n,q)$, *Trans. Amer. Math. Soc., ***358** (2006), 4935-4957.

They determined the Smith normals form of the incidence matrices of points and projective $(r-1)$-dimensional subspaces of $\operatorname{PG}(n,q)$ and of the
incidence matrices of points and $r$-dimensional aﬃne subspaces of $\operatorname{AG}(n,q)$ for all $n$, $r$, and arbitrary prime power $q$. If you want a quick summary of their results and other related known results in the language of design theory, a survey by the third author is available:

Q. Xian, Recent results on $p$-ranks and Smith normal forms of some $2$−$(v,k,\lambda)$ designs, *Contemp. Math.*, **381** (2005) 53-67.

I don't know much about the rank of the incidence matrix of an $S(t,k,v)$ over $\mathbb{F}_2$ when $t \geq 3$. But this paper studies the problem for $S(t,t+1,v)$s, and this one seems to give some results for $S(t,t+2,v)$s.

As for the generalization in the second direction, it's always full rank. For example, $N_2$ is ${{v}\choose{2}}$ binary row vectors of weight $1$ stacked together in which the column weights are uniformly $1$. And $N_3$ is the $\frac{v(v-1)}{6} \times \frac{v(v-1)}{6}$ identity matrix plus a bunch of row zero vectors underneath.

About the combination of the two types of generalizations, allowing $k > 3$ doesn't really change the situation; you always get a matrix of full rank. But increasing $t$ and/or $\lambda$ may lead to a nontrivial situation. For example, the classic rank formula for the $s$-subset vs. $t$-subset inclusion matrix by Wilson can be seen as an example of the generalization of this kind for the trivial $S(t,t,v)$ design:

R. M. Wilson, A diagonal form for the incidence matrices of $t$-subsets vs. $k$-subsets, *European J. Combin.*, **11** (1990), 609-615

But other than this, I haven't thought or heard about generalizations you asked.

A similar problem has been considered in the following papers on an application of modular representation theory:

A. Frumkin, A. Yakir, Rank of inclusion matrices and modular representation theory, *Israel J. Math.* **71** (1990), 309-320.

A. Yakir, Inclusion matrix of $k$ vs. $l$ affine subspaces and a permutation module of the general affine group, *J. Combin. Theory, Ser. A*, **63** (1993), 301–317.

Basically, they consider the rank of the "$s$-dimensional subspaces vs. $t$-dimensional subspaces" incidence matrices of projective/affine geometry over $\mathbb{F}_q$. So, for example, when $s = 1$, the problem reduces to the case of the standard incidence matrix $N$ of the corresponding $2$-designs. And the case when $s >1$ is the $s$-dimensional version of the problem you described.