Probably you should get a hold of a textbook on categorical logic for some of these questions. Of the references given in the Wikipedia article, the one by Lambek & Scott and the two with Lawvere as an author might be a good start. A study of these would probably help make the table in the nLab seem less "peculiar".

I see that Wouter Stekelenburg has posted an answer just as I started this, but it wouldn't hurt to amplify. The propositional connectives (on predicates of a given type $A$) are interpreted as operations on the poset of subobjects $Sub(A)$ (e.g., the pullback of two subobjects of $A$ is another subobject of $A$ which corresponds to the meet). Next, each morphism $f: B \to A$ in the category induces an inverse image operation $f^{-1}: Sub(A) \to Sub(B)$, formally defined by taking a pullback: if $i: U \to A$ is a monomorphism, then the pullback of the cospan $(f, i)$ is given by a pair of maps $(g: V \to U, j: V \to B)$ where one may show easily from general properties of pullbacks that $j$ is a monomorphism, thus giving a subobject of $B$. In the classical case where $f$ is a projection map $\pi: X \times A \to A$, you can think of the inverse image $\pi^{-1}: Sub(A) \to Sub(X \times A)$ as interpreting the operation which takes a predicate $\phi$ of type $A$ and forming a predicate of type $X \times A$ by harmlessly adjoining a variable of type $X$, e.g., by taking $\phi \wedge [x = x]$.

Then quantifiers $\exists_f$, $\forall_f$ are operations $Sub(B) \to Sub(A)$ that are left and right adjoint (respectively) to $f^{-1}: Sub(A) \to Sub(B)$. It might help to consider the classical case where $f$ is the projection $\pi: X \times A \to A$. Here the left adjoint $\exists_f$ takes a subobject of $X \times A$ (that interprets a predicate $\phi(x, a)$ of type $X \times A$) to one of type $A$, interpreting the predicate that is usually denoted $\exists_x \phi(x, a)$. (In other words, classical quantification has to do with adjoints to puling back along projection maps $\pi$.) The adjunction between $\exists_{\pi}$ and $\pi^{-1}$ corresponds to the inference rule

$$\exists_x \phi(x, a) \vdash \psi(a) \qquad \mathrm{iff} \qquad \phi(x, a) \vdash \psi(a) \wedge [x = x]$$

where the entailment symbols are interpreted as subobject inclusions in the appropriate posets $Sub(A)$, $Sub(X \times A)$. (Normally one isn't so fussy as to write $\psi(a) \wedge [x = x]$; one usually writes simply $\psi(a)$, but keeping in mind that $\psi$ is here being considered a predicate of type $X \times A$, so that the types match.)

Your guesses about the modalities are a little off. There are lots of places to read about categorical semantics of modal logic; one nice place (for S4 necessity) is in the paper by Awodey and Kishida. It might help first to be familiar with the topological semantics of "necessity" as a topological interior operation on power sets $\Box = \mathrm{int}: P(A) \to P(A)$, due originally to Tarski.