Here is a start at answering Question 1: there are indeed further relations between the $w_{2^k}$, or at least conditions on the $w_{2^k}$.

For example, consider the case where $H$ has multiplication which is null except for what is implied by the multiplication being unital. (For example, $H$ may be the cohomology of a suspension space.)

In this case, the Wu formula reduces to

$$Sq^i(w_j) = \binom{j-1}{i} w_{i+j}$$

So if the $w_{2^k}$'s are given, we are forced to define $w_{2^k + j'} = Sq^{j'} w_{2^k}$ for $0 \leq j' \leq 2^k-1$, which gives us the definition of each $w_j$. So now in the case where $j = 2^k+1$ and $1 \leq i \leq 2^k - 2$, the Wu formula stipulates that $Sq^i Sq^1 w_{2^k} = 0$. This is always the case for $i = 1$, but for all other $i$, the relation $Sq^i Sq^1 = 0$ does not hold in the Steenrod algebra, so I believe there are examples of $H$'s and $w_{2^k} \in H^{2^k}$ where this equation does not hold. So this is an example of some kind of further condition which $w_{2^k}$ may be required by the Wu formula to satisfy.

Here is a start at answering Question 1: there are indeed further relations between the $w_{2^k}$, or at least conditions on the $w_{2^k}$.

For example, consider the case where $H$ has multiplication which is null except for what is implied by the multiplication being unital. (For example, $H$ may be the cohomology of a suspension space.)

In this case, the Wu formula reduces to

$$Sq^i(w_j) = \binom{j-1}{i} w_{i+j}$$

So if the $w_{2^k}$'s are given, we are forced to define $w_{2^k + j'} = Sq^{j'} w_{2^k}$ for $0 \leq j' \leq 2^k-1$, which gives us the definition of each $w_j$. So now in the case where $j = 2^k+1$ and $1 \leq i \leq 2^k - 1$, the Wu formula stipulates that $Sq^i Sq^1 w_{2^k} = 0$. This is always the case for $i = 1$, but for all other $i$, the relation $Sq^i Sq^1 = 0$ does not hold in the Steenrod algebra, so I believe there are examples of $H$'s and $w_{2^k} \in H^{2^k}$ where this equation does not hold. So this is an example of some kind of further condition which $w_{2^k}$ may be required by the Wu formula to satisfy.