Briefly, descent is an analogue of taking quotients.

In the category of sets, we have the following facts:

1. an equivalence relations on a set is a relation $R \subseteq A \times A$ satisfying certain properties;

2. a quotient for an eq. rel. $R$ is a map $[-] : A \to Q$ that is universal among maps sending $R$ to equality;

3. we can construct a quotient for any equivalence relation (using equivalence classes);

4. for $x, y \in A$, we have $[x] = [y]$ if and only if $(x,y) \in R$;

5. every surjection $e : A \twoheadrightarrow A'$ is a quotient of $A$ by the relation “$e(x)=e(y)$”.

We can generalise these to arbitrary categories, ending up with the ingredients of a regular or exact category. The possibly-unexpected bit is the importance of property (4) — that elements become related in a quotient only if they were related by the given relation; this is not automatic from being a quotient, and is called an effective quotient. Lots of the ways we use quotients rely on effectivity (and not on anything else about the specific construction of the quotient).

The practical takehome is: If you can present an object as an effective quotient of a well-understood object, that can be very useful for understanding it. So it’s useful to have theorems for giving/identifying effective quotients, like fact (5) above.

Now since toposes are categorical objects, we have to generalise this to a 2-categorical version. When we “take a quotient” of an object of a 2-category, we want to “identify its elements/points/objects”, i.e. add new “isomorphisms”. But as ever, objects can be isomorphic via multiple different isomorphisms — equivalently, they can have non-trivial automorphisms. So when we “identify” them, we have to keep track of these extra automorphisms. So we want not just a “relation”, but extra data explaining what identifications should happen.

This is the idea of the truncated simplicial object that appears in the descent construction: it’s a generalisation of an equivalence relation. “Effective descent morphisms” are the analogue of effective quotients (and the category of descent data is a particular construction of such a quotient). And the motivation of the theorem is to give a tractable concrete presentation for an arbitrary topos, by expressing it as an “effective quotient” of a particularly nice and familiar kind of topos.

This tackles your questions 1–3. Questions 4 and 5 are a bit separate from these, and would be much better asked as separate questions.

Briefly, descent is an analogue of taking quotients.

In the category of sets, we have the following familiar facts:

1. an equivalence relations on a set is a relation $R \subseteq A \times A$ satisfying certain properties;

2. a quotient for an eq. rel. $R$ is a map $[-] : A \to Q$ that is universal among maps sending $R$ to equality;

3. we can construct a quotient for any equivalence relation (using equivalence classes);

4. for $x, y \in A$, we have $[x] = [y]$ if and only if $(x,y) \in R$;

5. every surjection $e : A \twoheadrightarrow A'$ is a quotient of $A$ by the relation “$e(x)=e(y)$”.

We can generalise these to arbitrary categories, ending up with the ingredients of a regular or exact category. The possibly-unexpected bit is the importance of property (4) — that elements become related in a quotient only if they were related by the given relation; this is not automatic from being a quotient, and is called an effective quotient. Lots of the ways we use quotients rely on effectivity (and not on anything else about the specific construction of the quotient).

The practical takehome is: If you can present an object as an effective quotient of a well-understood object, that can be very useful for understanding it. So it’s useful to have theorems for giving/identifying effective quotients, like fact (5) above.

Now since toposes are categorical objects, we have to generalise this to a 2-categorical version. When we “take a quotient” of an object of a 2-category, we want to “identify its elements/points/objects”, i.e. add new “isomorphisms”. But as ever, objects can be isomorphic via multiple different isomorphisms — equivalently, they can have non-trivial automorphisms. So when we “identify” them, we have to keep track of these extra automorphisms. So we want not just a “relation”, but extra data explaining what identifications should happen.

This is the idea of the truncated simplicial object that appears in the descent construction: it’s a generalisation of an equivalence relation. Effective descent morphisms of toposes are the analogue of effective quotients (and the category of descent data is a particular construction of such a quotient). And the motivation of the theorem is to give a tractable concrete presentation for an arbitrary topos, by expressing it as an “effective quotient” of a particularly nice and familiar kind of topos.

This tackles your questions 1–3. Questions 4 and 5 are a bit separate from these, and would be much better asked as separate questions.