Weierstrass Approximation for functions and (jacobian) derivatives simultaneously

Question

It is well known that polynomials are dense in $C_{[0,1]}$ the space of continuous functions on $[0,1]$. This is the celebrated Stone-Weierstrass theorem.

I am interested in generalizations of this that include approximations to the derivatives especially higher dimensional analogs using Jacobians. I am aware of one proof for the simple case of $C^1_{[0,1]}$

Case 1: If $f\in C^1_{[0,1]}$ then approximate $f'$ by some polynomial on $[0,1]$ via Weierstrass, and call it $q$. Write $p(x)=\int_0^1 q(x)dx+f(0)$. Then $$|f(x)-p(x)| \leq \int_0^1 |f'(x)-q(x)|dx$$ $$\leq \int_0^1 \|f'-q\|_\infty dx = \|f'-q\|_\infty,$$ which we can make as small as we want. The result follows.

Questions

1. Does this result hold for $C^1(\mathbb{R})$? The above method fails as is, since the last integral involves integrating over $(-\infty, \infty)$. Indeed the classic WAT is stated for compact intervals, so this might be a restriction that we can only get uniform approximations on compacts. How is this formulated precisely?
2. Does this result generalize to higher dimensions and using Jacobian matrices instead? i.e. can we prove it for $C^{1}(\mathbb{R}^n, \mathbb{R}^k)$? What distance should we use for measuring the approximation of Jacobians $D[p] \approx D[f]$? The Frobenius norm?

I apologize if these are a bit basic function approximation questions. I believe I saw the second question claimed in Rogers and Williams but without proof (I am trying to track down the passage to quote).

Edit 1 After a little thinking, I believe the method above can be generalized for higher dimensions as follows. Let $f \in C^1(\mathbb{R}^n, \mathbb{R})$. So its partials $\partial_i f$ are of class $C(\mathbb{R}^n, \mathbb{R})$, hence by WAT, on compact subsets, we can find a polynomial arbitrarily close to it in the uniform norm. Call it $q_i$. Now, since $f=\int \partial_i f dx^i+C$ where $C$ is a "constant" of integration not depending on the $i$-th coordinate $x^i$, if we let $p = \int q_i dx^i+C$ then $$|f(x)-p(x)|\leq \int |\partial_i f(x)-q_i(x)|dx^i$$ $$\leq C_0 \|\partial_i f-q_i\|_\infty,$$ where $C_0$ is some constant from integrating over the compact subset. Taking the supremum over $x$ in the compact subset yields the result. Then, coordinate-wise, $\nabla f(x)$ is arbitrarily approximated by $\nabla p(x)$. I think similarly this can be extended to $f\in C^1(\mathbb{R}^n, \mathbb{R}^K)$. Have I made any sloppy mistakes or is this about right?