Mathematically, there are two ways of "translating" "I like neither chocolate nor vanilla" (the two ways are logically equivalent, an instance of de Morgan's laws). You can write either:
$$\neg\bigl(\mathrm{like}(\mathrm{chocolate})\bigr)\ \&\ \neg\bigl(\mathrm{like}(\mathrm{vanilla})\bigr)$$
(that is, "I don't like chocolate and I don't like vanilla") or as
$$\neg\bigl(\mathrm{like}(\mathrm{chocolate})\ \vee\ \mathrm{like}(\mathrm{vanilla})\bigr).$$
The two are equivalent, because $\neg(P\vee Q) \equiv (\neg P)\&(\neg Q)$ (this is one of De Morgan's Laws: for "P or Q" to be false, you need both P to be false and Q to be false).
The second sentence is similar: "Neither e nor a is to the right of c and to the left of b".
You mention you are confused about it; I think it's just a confusion of parsing (it is a bit awkwardly constructed). What is says is "Neither e nor a satisfy xxxx". That is: "e does not satisfy xxxx, and a does not satisfy xxxx". What is this xxxx? It is the condition "is to the right of c and to the left of b".
So the sentence is the same as "neither
(Note that "e is not to the right of c and to the left of b" means that either e is not to the right of c, or e is not to the left of b, or both; same with a).
If you wanted to translate it in the same manner as the first translation above, with conjunction, you would have:
$$\Bigl(\neg\bigl(\mathrm{RightOf}(e,c)\&\mathrm{LeftOf}(e,b)\bigr) \Bigr) \& \Bigl(\neg\bigl(\mathrm{RightOf}(a,c)\&\mathrm{LeftOf}(e,b)\bigr)\Bigr).$$
But you can equally as well use the model of the second translation above, to get the equivalent statement:
$$\neg\Biggl( \bigl(\mathrm{RightOf}(e,c)\&\mathrm{LeftOf}(e,b)\bigr) \vee \bigl(\mathrm{Rightof}(a,c)\&\mathrm{LeftOf}(a,b)\bigr)\Biggr).$$
So now you see the disjunction in the second.
The reason why the first "translation" of this phrase might be better than the second is that the mix of conjunctions and disjunctions ( "(a and b) or (x and y)" ) is usually a little harder to parse than a sequence of conjunctions, even with negations ( "not(a and b) and not(x and y)"). So the latter form is slightly prefered for parsing reasons, but mathematically they are equivalent. The parsing problem does not show up in the vanilla/chocolate example, because there are no conjunctions hiding inside the clauses and complicating the parsing.
Best Answer
First, you suggest propositions, but you need to define them explicitly. We will need three propositions:
$P:$ Pliny is happy.
$Q:$ Quintilian is happy.
$R:$ Rufina is happy.
So you are given:
$(1)\;$ Pliny and Quintilian are happy.
$(2)\;$ If Pliny is happy, then Quintilian is happy or Rafina is happy.
$(3)\;$ If neither Pliny nor Quintilian is happy, then Rafina is happy.
$\;P \land Q$
$\;P\to (Q\lor R)$
$\;(\lnot P \land \lnot Q)\to R$