Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have high rank?

This is not a complete answer, it requires some literature search. But I believe still that the answer to this question is negative and the proper setting of this question is within singularity theory and things like stable singularities.

Namely, if you go to the book

Singularity Theory I. Author(s): V. I. Arnold, V. V. Goryunov, O. V. Lyashko, V. A. Vasil’ev (auth.) Series: Encyclopedia of Mathematical Sciences 6

and look on page 160, there is a theorem, attributed to Porteuos (Porteous, I.R.: The second-order decomposition of $\Sigma^2$ Topology 11, 325-334 (1972).),

Theorem. A generic map $f: N^n\to P^n$ can be written, in suitable coordinates in a neighborhood of an elliptic [resp. hyperbolic] singular point 0 as

$$y_1=x_1,\ldots, y_{n-2}=x_{n-2}, \, y_{n-1}=x_{n-1}x_n,\,y_n=x_{n-1}^2-x_n^2+x_{n-3}x_{n-1}+x_{n-2}x_n$$ [resp $y_n=x_{n-1}^2+x_n^2+...$]

This formula makes sense starting from $n=4$. Note that the rank of the map at $(0,\ldots, 0)$ is $n-2$. So I interpret this theorem that the map $\mathbb R^4\to \mathbb R^4$ given by the above formula can not be perturbed in such a way so to get rid of a point where the rank of the differential equals $2$. This result is about small smooth perturbations, but I doubt that involving the functional space $W^{1,2}$ will make any difference.

ADDED: Easier question.

Concerning the second part of the question, it should follow from the result called "Strong tranversality Theorem" in the book of Arnol'd, Varchenko, Gussein-Zade, page 38 "Singularities of differentiable maps, volume 1"

Theorem.(Thom) Let $M^n$ be a closed manifold and $C$ a closed submanifold of the space of jets $J^k(M,N)$. Then the set of maps $f:M\to N$ whose $k$-jet extensions are transversal to $C$ is an open everywhere dense set in the set of all smooth maps from $M$ to $N$.

In the case under consideration it is enough to consider $1$-jets, since we are only interested in the condition on the rank of the differential. Now this theorem will give you codimension 4, as was noticed in the comments.