Questions about differential forms Let $\omega$ be a $k$-form on $M \subseteq \mathbb R^n$ a $d$-dimensional submanifold of $\mathbb R^n$ and $k,d \le n$.
1) Can I only integrate $\omega$ over $M$ if k = d? Why?
2) Let $\omega$ be a $1$-form and $\alpha: \mathbb R  \to \mathbb R^n$ a curve. If I integrate $\omega$ along $\alpha$, I compute the scalar product of the
vectors (given by $\omega$) on $\alpha$ and the tangential vector that spans the tangential space of $\alpha$. (This is not quite precise because $\omega$ maps into the dual space of $\mathbb R^n$, but this is just supposed to visualize the concept of this integral). Does the integration of a $k$-form ($k$ > 1) over $M$, in a way, generalize the scalar product? What kind of integration is this?
4) What is the connection between the exterior derivative and the 'classical' higher-dimensional derivative?
Note that a submanifold of $\mathbb R^n$ is way more than just a subset of 
$\mathbb R^n$. 
EDIT1: I withdraw question 3): "Why does $\omega$ assign each $p∈M$ a different multilinear form?", because it follows basically by definition.
EDIT2: I think that it would have been smarter to let $\alpha$ go from $[a,b] \to \mathbb R^n$

The answer to your first question comes from measure theory . The way integration is defined is  only possible for the forms which are of the same  length as the dimension.  Also , for me i think integration as a pairing of k forms and n-k forms .Thus then it makes sense only for the top forms.
For the third question , i think at first $\omega$ a k form as a section of the cotangent bundle of    k th exterior algebra of $\mathbb{R}^n$ which is the dual of the tangent bundle. If you are familiar with this definition , then it completely makes sense to have $\omega(p)$ as a multilinear form (in particular alternating) for each p .
