Continuity of the distributional jacobian

If $u \in C^2(\overline\Omega;\mathbb R^n)$ we have $$ \int_{\Omega} \phi \det \mathrm{D}u \,\mathrm{d}x = \int_{\Omega} \phi \,\frac1n \operatorname{div}(\operatorname{adj} \mathrm{D}u \cdot u) = - \frac1n \int_{\Omega} \mathrm{D} \phi \cdot\operatorname{adj} \mathrm{D}u \cdot u\,\mathrm{d}x\tag{$\dagger$} $$ for any $\varphi \in C^{\infty}_c(\Omega)$, which is the motivation to define the distributional Jacobian $\operatorname{Det} \mathrm{D}u$ using the right-hand side.

Q1. If $u \in W^{1,p}(\Omega;\mathbb R^n)$ with $p \geq n$, then in particular $\mathrm{D}u \in L^p(\Omega;\mathbb R^{n\times n})$, so $\det \mathrm{D}u$ is a well-defined $L^1$ function on $\Omega$. So by approximating $u$ by smooth maps $u_k$ in $W^{1,p}(\Omega;\mathbb R^n)$, we can pass to the limit in $(\dagger)$ to conclude that $\det \mathrm{D}u = \operatorname{Det} \mathrm{D}u$. This is a standard density argument, and positively answers your first question.

Q2. If $u(x) = \frac x{|x|}$ is the vortex map considered in $\Omega = B_1$, you can use the same strategy to compute $\operatorname{Det} u$. Here $u$ no longer lies in $W^{1,n}(B_1;\mathbb R^n)$, so $\det \mathrm{D} u$ no longer a well-defined integrable function $B_1$, so we need to understand it distributionally.

However if $u_k$ is a sequence of functions in $C^2(\overline\Omega;\mathbb R^n)$ such that $u_k \to u$ uniformly and also in $W^{1,p}$ for some $p \in [n-1,n)$, then we can pass to the limit in $(\dagger)$ with $u_k$ to show that \begin{align*} \langle \operatorname{Det} u, \phi \rangle &= -\frac1n \int_{\Omega} \mathrm{D} \phi \operatorname{adj} \mathrm{D}u \cdot u\,\mathrm{d}x \\ &= -\lim_{k \to \infty} \frac1n \int_{\Omega} \mathrm{D} \phi \cdot\operatorname{adj} \mathrm{D}u_k \cdot u_k\,\mathrm{d}x \\ &= \lim_{k \to \infty}\int_{\Omega} \phi \det \mathrm{D} u_k \,\mathrm{d}x. \end{align*} So we see that if we approximate $u$ by nice functions, the distributional Jacobian is a suitable limit of the approximating Jacobians.

To actually compute $\operatorname{Det} \mathrm{D}u$ is more involved, but one just need to use a suitable approximation. Typically one takes a radial cutoff $\eta_{\varepsilon}(|x|)$ which vanishes near the origin, and take $u_{\varepsilon}(x ) = \eta_{\varepsilon}(|x|) u(x)$. I don't know of a reference that does this explicitly, but one has to compute $\det \mathrm{D}u_{\varepsilon}$ and to see what happens as $\varepsilon \to 0$.