The answer presented here is copied from the paper [2]. Many similar results (sometimes with more complicated proofs) can be found in [1].
Let the space $L^{k,p}$ be defined by:
$$
L^{k,p}(\mathbb{R}^n)=\{ f\in \mathcal{D}'(\mathbb{R}^n):\, \nabla^kf\in L^p(\mathbb{R}^n)\},
\quad
\Vert f\Vert_{L^{k,p}}=\Vert \nabla^kf\Vert_p.
$$

**Theorem** (Theorem 4 in [2]). *Let $1\leq p<\infty$ and $k=1,2,\ldots$ Then $\mathcal{D}(\mathbb{R}^n)$ is dense in $L^{k,p}$ if and only if
$n>1$ or $p>1$.*

Here density is with respect to the seminorm $\Vert \nabla^kf\Vert_p$. That is for any $f\in L^{k,p}(\mathbb{R}^n)$ there is a sequence $f_i\in \mathcal{D}(\mathbb{R}^n)$ such that $\Vert \nabla^k(f-f_i)\Vert_p\to 0$.
Therefore the space coincides with the completion that is described in the question i.e.,
$\mathring {L^k_p}(\mathbb{R}^n)=L^{k,p}(\mathbb{R}^n)$.
Formally, the spaces $L^{k,p}$ are defined as distributions and only their $k$-th order derivatives are functions. However we have
$$
L^{k,p}(\mathbb{R}^n)\subset W^{k,p}_{\rm loc}(\mathbb{R}^n)=\{f\in L^p_{\rm loc}:\, \nabla^\ell f\in L^p_{\rm loc} \text{ for all $0\leq\ell\leq k$\}}.
$$
This is a classical result, but a short and self-contained proof is given in this post https://mathoverflow.net/a/296464/121665.

**This gives an answer to questions 2 and a partial answer to question 1:**

$$ \mathring {L^k_p}(\mathbb{R}^n)=L^{k,p}(\mathbb{R}^n)\subset
W^{k,p}_{\rm loc}(\mathbb{R}^n). $$

Clearly $u$ is a function and $\nabla^k u$ defined as a limit of approximations (as in question 2) is a weak derivative of $u$.

**So how does a typical element of $L^{k,p}$ look like?**

If $\Omega$ is a bounded domain, then by the Poincare inequality
$\Vert u\Vert_{W^{k,p}}\leq \Vert \nabla^k u\Vert_{p}$ for
$u\in\mathcal{D}(\mathbb{R}^n)$. This gives

If $\Omega$ is a bounded domain, then $\mathring
{L^k_p}(\Omega)=W^{k,p}_0(\Omega)$.

A precise description of the space $\mathring {L^k_p}(\mathbb{R}^n)$ is available when $kp<n$.

If $kp<n$ or $k=n$, $p=1$ we define the homogeneous Sobolev space by
$$
\mathring W^{k,p}(\mathbb{R}^n)=\left\{f:\, \nabla^\ell f\in L^{p_{\ell}^*}(\mathbb{R}^n),\ \
p_\ell^*=\frac{np}{n-(k-\ell)p},\ \ \ 0\leq\ell\leq k\right\}.
$$
This space is equipped with the norm
$$
\Vert f\Vert_{\mathring W^{k,p}} =\sum_{\ell=0}^k \Vert\nabla^\ell f\Vert_{p_\ell^*}.
$$

It follows from the Sobolev embedding that $W^{k,p}\subset \mathring W^{k,p}$, but the later space is larger. The next result provides a **complete
answer to question 1**
in the case when $kp<n$ and $1\leq p<\infty$.

**Theorem.** (Theorem 5 in [2]). Let $kp<n$ and $1\leq p<\infty$. The for every $f\in L^{k,p}(\mathbb{R}^n)$ there exists a unique
polynomial $P^{k-1}f\in \mathcal{P}^{k-1}$ of degree $\leq k-1$ such that $f-P^{k-1}f\in
\mathring W^{k,p}(\mathbb{R}^n)$. Morover $$ \Vert f- P^{k-1}f\Vert_{
\mathring W^{k,p}}\leq C\Vert\nabla^k f\Vert_p. $$

In other words $L^{k,p}= \mathring W^{k,p}+\mathcal{P}^{k-1}$.

For other related results, see [1], [2] and [3].

[1] O. V. Besov, V. P. Ilʹin, S. M. Nikolʹskiĭ, *Integral representations of functions and imbedding theorems.* Vol. I & II. Scripta Series in Mathematics. Edited by Mitchell H. Taibleson. V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1979.

[2] P. Hajłasz, A. Kałamajska, Polynomial asymptotics and approximation of Sobolev functions, *Studia Math.* 113 (1995), 55-64.

[3] P. J. Rabier,
$L^p$ regularity of homogeneous elliptic differential operators with constant coefficients on $\mathbb{R}^N$.
*Rev. Mat. Iberoam.* 34 (2018), 423–454.

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