Questions about tangent and cotangent bundle on schemes

In differential geometry, for a smooth manifold $M$ we have the definition of the tangent bundle and the cotangent bundle and then $k$-forms are defined to be (smooth) sections of the $k$-exterior power of the latter.

In algebraic geometry, say $X$ is the affine space $\mathbb{A}_k^n=\operatorname{Spec}(k[x_1,\ldots,x_n])$, we can talk about the Zariski tangent space at a point $x$ which is by definition $T_xX=(\mathfrak{m}_x/\mathfrak{m}_x^2)^*$ ($\mathfrak{m}_x$ is the maximal ideal $\mathcal{O}_{X,x}$). I have read that this definition is not the best one unless the point $x$ is $k$-rational point. For other points there exists the concept of relative tangent space (see for example https://www.math.ucdavis.edu/~blnli/buildings/bag.pd, page 151).

1) My first question is: Is there exists the notion of tangent bundle on $X$ as a sheaf?

Comming back to differential geometry, I have read on the wikipedia a nice definition of the cotangent bundle: let $M$ be a smooth manifold and let $\mathcal{I}$ be the sheaf of of germs of smooth functions on M×M which vanish on the diagonal $\Delta(M)\subseteq M\times M$, and define the contangent bundle $T^*M$ as the sheaf $\Delta^*(\mathcal{I}/\mathcal{I}^2)$ (where $\Delta:M\rightarrow M\times M$ is the diagonal map). This makes me suspect that a possible definition of the contangent space on a separated scheme $X$ over a field $k$ will be the same.

2) So my second question is: is this definition of cotangent bundle correct in algebraic geometry?.

3) (More urgent) I make these questions because while reading a paper I found this part: "Let $h:Y\rightarrow X=\operatorname{Spec}(k[x_1,\ldots,x_n])$ be a morphism of integral schemes and consider the divisor of $h^*(dx_1\wedge\ldots\wedge dx_n)$ ... ". I would appreciate a lot if you help me to interpret this part. What is exactly $dx_1\wedge\ldots \wedge dx_n$?, why can we obtain a divisor from $h^*(dx_1\wedge\ldots\wedge dx_n)$?.

(1) Yes, definitely; and you'll be able to check that the fiber of $\Omega_{X/k}$ t a $k$-rational point is the old cotangent space.
(3) The $k[x_1, \dots, x_n]$-module $\Omega_{k[x_1, \dots, x_n]/k}$ is free on the symbols $dx_i$, and $dx_1 \wedge \cdots \wedge dx_n$ is thus a section of the top exterior power, which corresponds to a invertible sheaf on $\mathbb{A}^n_k$ (this has to be globally trivial, but that doesn't matter for what I'm going to say). Pull back the section to get a section of the invertible sheaf $h^*\Omega_{\mathbb{A}^n_k/k}$ and take the divisor of that.
In general from a morphism of $k$-schemes $f\colon X \to Y$ we do get a pull-back $f^*\Omega_{Y/k} \to \Omega_{X/k}$ that you would expect from differential geometry, but unless the source in your example is smooth of dimension $n$ then I don't see how to get a divisor out of this.