Counting functions on the Boolean cube

Which subsets of the vertices of $\mathbb {B}^{4}$ correspond to the following Boolean functions:

Call the tuples $\alpha =\left(\alpha_{1}, \ldots , \alpha_{n}\right) \in \mathbb {B}^{n}$ and $\beta =\left(\beta_{1}, \ldots , \beta_{n}\right) \in \mathbb {B}^{n}$ neighboring if they occur in adjacent rows of the table for a function of $n$ variables, that is, they represent the binary expansions of numbers $i_{\alpha }$ and $i_{\beta }$ for which $\left|i_{\alpha }-i_{\beta }\right|=1$.

Find the number of functions in $\mathcal{P}_{n}$ that take, on every pair of neighboring tuples,

Call the tuples $\alpha =\left(\alpha_{1}, \ldots , \alpha_{n}\right) \in \mathbb {B}^{n}$ and $\beta =\left(\beta_{1}, \ldots , \beta_{n}\right) \in \mathbb {B}^{n}$ opposite if $\alpha_{i}+\beta_{i}=1$ for every $i$ (in other words, $\alpha_{i}=1$ if and only if $\beta_{i}=0$).

Find the number of functions in $\mathcal{P}_{n}$ that take different values on every pair of opposite tuples.

Let $n=2 k$. Call a tuple $\alpha =\left(\alpha_{1}, \ldots , \alpha_{n}\right) \in \mathbb {B}^{n}$ paired if $\alpha_{i}=\alpha_{k+i}$ for all $i=1, \ldots , k$, that is $\alpha =\alpha^{\prime } \alpha^{\prime }$ for some tuple $\alpha^{\prime }$ of size $k$. Find the number of functions in $\mathcal{P}_{n}$ that take the same value on all paired tuples.

Which subsets of the vertices of $\mathbb {B}^{4}$ correspond to the following Boolean functions: