The Generalized Hermitian Gell-Mann Matrices (in dimension $d$ ) consist of the $h_k^d$, where $1\leq k \leq d$, and the $f_{k,j}^d$, where $1\leq k, j\leq d$.
There should be $2^d -1$ matrices in dimension $d$. But here there is only $d-1$ from the $h$ type and maybe $d^2 -d$ of the $f$ type?
No, there's $d^2 - 1$ matrices in dimension $d$, not $2^d-1$. Here is a prescription for generating the matrices.
The $d^2-1$ matrices can be categorized as follows:
In total this becomes:
$$\tag{1} \frac{d(d-1)}{2} + \frac{d(d-1)}{2} + (d-1) = d^2-1. $$
This decomposition is also given explicitly in Eqs. 3-5 of this paper.
In the Wikipedia article that you referenced, the identity matrix is included, so there would be $d^2$ matrices, not just $d^2 - 1$.