The idea behind a "collection" is simply a notion of a bunch of mathematical objects which are collected into one big pile. Think of it as a big bin full of trash, diamonds and empty bottles of beer, it doesn't have to make sense what is in this collection, it's just a collection.
One of the problem to explain these things to people who are not mathematicians (or trying to "outsmart a set theorist", as I ran into several of those) is that the notion of a collection is not fully formal unless you already know what sets and class are, and even then it's not exactly what we mean.
Let me start over now. Doing mathematics we often have an idea of an object that we wish to represent formally, this is a notion. We then write axioms to describe this notion and try to see if these axioms are self-contradictory. If they are not (or if we couldn't prove that they are) we begin working with them and they become a definition. Mathematicians are guided by the notion but they work with the definition. Rarely the notion and the definition coincide, and you have a mathematical object which is exactly what our [the mathematicians] intuition tells us it should be.
In this case, a collection is a notion of something that we can talk about, like a mystery bag. We might know that all the things inside this mystery bag are apples, but we don't know which kind; we might know they are all Granny Smith, but we cannot guarantee that none of them is rotten. A collection is just like that. We can either know something about its elements or we don't, but we know that it has some.
Mathematician began by describing these collections and calling them sets, they did that in a relatively naive way, and they described the axioms in a rather naive way. To the non-mathematician (and to most of the non-set theorists) everything is still a set, and we can always assume that there is a set theorist that assured that for what we need this is true. Indeed, if we just wanted to discuss the real numbers, there is no worry at all we can assume everything we work with is a set.
This naive belief can be expressed as every collection is a set. It turned out that some collections cannot be sets, this was expressed via several paradoxes, Cantor's paradox; Russell's paradox; and other paradoxes. The exact meaning is that if we use that particular axiomatic description of "what is a set" then we can derive from it contradiction, which is to say that these axioms are inconsistent.
After this happened several people began working on ways to eliminate this problem. One method in common was to limit the way we can generate collections which are sets. This means that you can no longer derive such contradiction within the theory, namely you cannot prove that such collection even exists - or rather you can prove it doesn't.
The common set theory nowadays called ZFC (named after Zermelo and Fraenkel, the C denotes the axiom of choice) is relatively close to the naive way from which set theory emerges, and it still allows us to define collections which are not sets, though, for example "the collection of all sets". These collections are called classes, or rather proper classes.
What is definable? This is a whole story altogether, but essentially it means that we can describe it with a single formula (perhaps with parameters) of one free variable. "$x$ is taller than 1.68m" is an example to such formula, and it defines the class of all people taller than said height.
So in ZFC we may define a collection which is not a set, like the collection of all the singletons, or the collection of all sets. These are not sets because they are too big, in some sense, to be sets, but they are classes, proper classes. We can talk about collections which are not definable but that requires a lot more background in logic and set theory to get into.
To sum up
Classes are collections which can be defined, sets are particular classes which are relatively small and there are classes which are not sets. Collections is a notion which is expressed via both these mathematical objects, but need not be well-defined otherwise.
Of course when we say defined we mean in the context of a theory, e.g. ZFC. In this sense, sets are things which "really exist" whereas classes are collections we can talk about despite their possible nonexistence.
One last thing remains, families. Well, as you remarked families are functions. But functions are sets, so families are sets. We can make a slight adjustment to this, and we can, in fact, talk about class functions, and an index which is not a set but a proper class. We, therefore, can talk about families which are classes.
Generally, speaking, if so, a family is a correspondence from one collection into another which uses one collection as indices for elements from another collection.
To read more
What is the difference between a class and a set?
Why is "the set of all sets" a paradox, in layman's terms?
Yes; it actually means that, there is some $N$ such that, for all $n>N$ we have $A_{N}=A_{n}$ - that is, all the sets are the same. This is basically what you're thought that $A_1$ has only finitely many subsets amounts to. The proof of this is basically the same as saying that the sequence of cardinalities $$|A_1|\geq |A_2|\geq |A_3|\geq \ldots$$
will be constant for large enough $N$ - that is, you cannot have an infinite descending sequence of natural numbers. (This is provable by induction, and essentially the idea induction captures in the first place)
Then, you can safely say that, for that $N$, we have $A_{i}\supseteq A_N$ for all $i$. Thus, the intersection of all of the terms of the sequence (which is what the $\infty$ in the intersection is trying to say) must contain $A_N$, since all the terms in the intersection do. However, it is also contained in $A_N$, since $A_N$ is a member of the set over which the intersection is being taken. Thus, it equals $A_N$, which is a finite, non-empty set.
Best Answer
I think that you are radically over-analysing the language. A "collection" of (sub)sets, a "family" of (sub)sets, and a "set" of (sub)sets are all referring to the same kind of object: a set whose elements are subsets of some other set. Note, even, that the Wikipedia disambiguation page for "collection" links to "set" and "family of sets". Wikipedia does seem to make some distinction between a "family of sets" and the other terms, but this is likely highly context-dependent, and such a distinction probably wouldn't matter to someone who is not an expert in set theory.
As examples:
I suspect that the terms "family" and "collection" exist as a form of disambiguation. "A set of sets" is somewhat awkward to say, and in verbal communication I can see where it would be useful to distinguish between the underlying set, and the set which contains subsets of the underlying set (e.g. we can refer to a topology as a "collection", then talk about element of the collection and subsets of the underlying set).