The relation between the inner product of vectors and the interior product is that if you have a metric tensor (and thus a canonical relation between vectors and covectors = $1$-forms), the inner product of two vectors is the interior product of one of the vectors and the $1$-form associated with the other one. That is, if $g$ is the metric tensor, then the inner product of the vectors $v$ and $w$ is $g(v,w)$, and the $1$-form $\omega$ associated with $v$ is defined by $\omega(w)=g(v,w)$. Then it is obvious that $\iota_w(\omega) = \omega(w) = g(v,w)$.
The outer product is, as noted in answer to the other question you referred to, related to the tensor product. Indeed, if we associate row vectors with $1$-forms and column vectors with vectors, then we can write (using Einstein summation convention) the outer product of the vector $w = w^ie_i$ and the $1$-form $\omega = \omega_i\,e^i$ as the $(1,1)$ tensor $M = w^i\omega_j e_i\otimes e^j$ which describes an object that maps vectors to vectors. Its relation to the inner product is that you get the inner product of $w$ and $\omega$ by contracting the two indices of $M$ (which in the language of matrices corresponds to the trace of $M$).
The exterior product is related to the tensor product in that the exterior product of two forms (a form is a skew-symmetric tensor of type $(0,p)$) is just the antisymmetrization of the tensor product.
The cross product is a speciality of the three-dimensional space; here the space of $2$-forms has the same dimension as the space of $1$-forms; indeed, given a metric, the hodge star maps between them. Since the metric also allows to associate vectors and $1$-forms, you can define the cross product of $v$ and $w$ by the following procedure: Determine the $1$-forms corresponding to $v$ and $w$, calculate their exterior product (which is a $2$-form), apply the Hodge star to the result (which, given that we are in three dimensions, again results in a $1$-form), and finally determine the vector corresponding to that $1$-form.
Dot-products and cross-products are products between two like things, that is: a vector, and another vector. In a matrix-vector product, the matrix and vectors are two very different things. So, a matrix-vector product cannot rightly be called either a dot-product or a cross-product.
That being said, the matrix-vector product is closely related to the dot product. In particular: suppose that $A$ is a matrix with row-vectors $A_1,\dots,A_n$, and $b$ is a column vector. Then the product $Ab$ will be the column vector with entries $(A_1 \cdot b,\cdots,A_n \cdot b)$.
Moreover: given two column vectors $u$ and $v$, their dot-product is the same as the matrix product $u^Tv$, where $T$ here means the transpose. In this sense, we might consider the dot-product to be a kind of matrix product, but the reverse is not generally true.
Best Answer
Cross product is much more related to exterior product which is in fact a far going generalization.
Outer product is a matricial description of tensor product of two vectors.