$\newcommand{\Reals}{\mathbf{R}}\newcommand{\eps}{\varepsilon}$I'll briefly address the third question (the purpose of considering the dual space), since that's the most nebulous, subtle, and important in differential geometry.

In the setting of vector calculus (on a smooth $n$-dimensional manifold $M$), you have notions of local coordinates, change of coordinates, smooth functions, and smooth curves.

A smooth curve $\gamma:(-\eps, \eps) \to M$ defines a *tangent vector* $v = \gamma'(0)$ at the point $p = \gamma(0)$. The set of all tangent vectors at $p$ turns out to be an $n$-dimensional vector space, the *tangent space* $T_{p}M$.

A smooth, real-valued function $f:M \to \Reals$ defines a *differential* $df$. The "value" $df(p):T_{p}M \to \Reals$, which satisfies
$$
\frac{d}{dt}\bigg|_{t = 0} (f \circ \gamma)(t) = \bigl[df(p)\bigr](v),
$$
is *linear* in $v$, and therefore constitutes a *covector*, i.e., an element of the vector space dual to $T_{p}M$.

A finite-dimensional real vector space $V$ and its dual space $V^{*}$ have the same dimension, and so are isomorphic. However, there is no basis-independent way to define an isomorphism of $V$ with $V^{*}$. Consequently, vector-valued functions on $M$ (a.k.a. *vector fields*) and covector-valued functions on $M$ (a.k.a. *differential one-forms*) are distinct types of entities. For instance:

If $\phi:M \to N$ is a smooth map of smooth manifolds, a one-form on $N$ pulls back (in a coordinate-invariant way) to a one-form on $M$, but a vector field on $N$ does not pull back in a coordinate-invariant way to a vector field on $M$. (A tangent *vector* on $M$ naturally pushes forward to a tangent vector on $N$, but a vector *field* on $M$ does not naturally push forward to a vector field on $N$ in general.)

If $f:M \to \Reals$ is smooth, the differential $df$ depends only on the smooth structure of $M$, but there is no natural way to define a gradient vector field of $f$.

A smooth curve in $M$ has a natural notion of velocity (first derivative), but has no natural notion of acceleration (second derivative).

This list is in no way comprehensive. All such items come down to the chain rule (or, in fancy language, to the transition functions for the tangent and cotangent bundles of $M$), see also Dual space and covectors: force, work and energy for detailed computations along similar lines.

The second and third items are often avoided by working on a *Riemannian* manifold, equipped with a Riemannian metric $g$ (which can be used to define an isomorphism between vectors and covectors) and its Levi-Civita connection (which defines *covariant differentiation*, such as acceleration of a curve).

The fact that a function has a gradient field and a smooth curve has an acceleration in elementary vector calculus secretly relies on the Euclidean metric on Cartesian space.

*Tensor Analysis on Manifolds* by Bishop and Goldberg, and *Comprehensive Introduction to Differential Geometry, Volume I* by Spivak, are excellent sources for further reading. (There are surely more recent treatments, as well.)