How to prove that there is an entourage $V$ such that $V(B) \cap A = \emptyset$ in a uniform space when A is compact and B is closed?

$A$ and $B$ should, of course, be assumed to be disjoint; otherwise the conclusion obviously fails.

For each $a\in A$, we have an open neighborhood of $a$ disjoint from $B$, because $B$ is closed, and this neighborhood can be taken to be of the form $U_a(a)$for some entourage $U_a$. Choose symmetric entourages $V_a$ with $V_a\circ V_a\subseteq U_a$. Since $A$ is compact and is covered by the open sets $V_a(a)$ (for $a\in A$), it is covered by finitely many of these, say the ones corresponding to $a\in F$, where $F$ is a certain finite subset of $A$.

Now let $V=\bigcap_{a\in F}V_a$; since $F$ is finite, this is an entourage. I'll show that $V(A)$ is disjoint from $B$. It suffices to show that $V(x)$ is disjoint from $B$ for all $x\in A$, since $V(A)$ is the union of these. So consider any $x\in A$ and note that, by our choice of $F$, $x\in V_a(a)$ for some $a\in F$. Fix such an $a$. Since $V$ is symmetric and $\subseteq V_a$, we have $V(x)\subseteq V_a(x)\subseteq V_a(V_a(a))\subseteq U_a(a)$, which is disjoint from $B$ as required.

2.2.1 is easy to see as $$2^X\setminus \{E \in 2^X\mid E \subseteq A\} = \langle X, X\setminus A\rangle$$

and so the complement of the set is basic Vietoris open. (we use the $X$ so that the union part is trivial and the only demand to be in the basic set is to intersect the complement of $A$, the complementary condition to be a subset of $A$).

Similarly for 2.2.2:

$$2^X\setminus \{E \in 2^X\mid E \cap A \neq \emptyset\} = \{E \in 2^X\mid E \subseteq X\setminus A\}=\langle X\setminus A\rangle$$

so the complement of the set of interest is basic Vietoris open.

For uniform topologies we have to reason a bit differently, of course, with entourages or local bases instead.

Suppose $B \notin \{E \in 2^X\mid E \subseteq A\}$. This means that there is some $b \in B$ with $b \notin A$. So we find an entourage $V \in \mathcal{U}$ so that $V(b) \cap A = \emptyset$. The generating set $V(B)$ for $2^{\mathcal{U}}$, introduced in the paper in definition 1.6 (he only introduces the other sets you mentioned for the uniformity later on), then also misses $\{E \in 2^X\mid E \subseteq A\}$, showing closedness.

The final one is similar, and close to what you tried, using the lemma that Andreas Blass already ably showed in another answer and which I won't repeat.