Yes, those are equal and moreover, you do not even need the Jacobian from $x^{\mu}$ to $x^{'\mu}$, because of the fact that the authors, when they write $dx^4$, they actually mean the invariant integration measure $\sqrt{|g|}dx^0dx^1dx^2dx^3$.

You can verify yourself that the aforementioned integration measure is invariant under general coordinate transformations.

Yes, those are equal and moreover, you do not even need the Jacobian from $x^{\mu}$ to $x^{'\mu}$, because of the fact that the authors, when they write $dx^4$, they actually mean the invariant integration measure $\sqrt{|g|}dx^0dx^1dx^2dx^3$.

You can verify yourself that the aforementioned integration measure is invariant under general coordinate transformations.

So, the following two equations hold separately:

1. $\mathcal{L}' \Big(\phi'(x^{'\mu}),\partial'_{\rho}\phi(x^{'\mu}),x^{'\mu}\Big)= \mathcal{L} \Big(\phi(x^{\mu}),\partial_{\rho}\phi(x^{\mu}),x^{\mu}\Big)$

2. $dx^{'4}=\sqrt{g}dx^{'0}dx^{'1}dx^{'2}dx^{'3}=\sqrt{g}dx^{0}dx^{1}dx^{2}dx^{3}=dx^4$ where $\sqrt{g}=\sqrt{|\text{det}(g)|}$