I have not read it myself, however I got a good recommendation from one of my teachers -
Azriel Levy's Basic Set Theory.
Jech's Set Theory is a great book but I think it is indeed slightly too advanced, he writes that the first part contains full proofs (I only read chapters from the second parts, in which proofs are many times sketched out and the details are left for the reader). Once you've got the basic theorems down, one might also check The Handbook Of Set Theory written by an ensemble of competent writers, for more specific topics.
Does the mathematical definition of a set specify/imply that its elements be unique?
Yes.
The Set Theory Wikipedia page does not use the term "unique" or "distinct" in reference to set elements.
No, but it does say this:
"Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or element) of A, the notation o ∈ A is used."
I interpret this as: To be an object in the universe is to be unique. For every thing, it is the only thing that is itself and either an object $o$ is a member of a set $A$.... or it isn't.
Now a set has nothing about how order the things or how you pick them out or list them. And if $o$ is in the set $A$, it doesn't matter if when asked to describe the elements of $A$ I mention $o$ first, or last, or $53$rd and if I say "$A$ has $o$ and it has $t$ and $s$ and it has $o$, did I mention $o$ already, and it has $q$ and $z$ and $o$ and $m$ and $o$ and $o$ and, gee I'm mentioning $o$ a lot, and ..." The fact remains either $o$ is in the set or not. Those are the only options.
So if a set is $\{1,2,3,4\}$ that doesn't mean we can't list it as $\{4,3,4,1,2,4\}$. In fact consider $\mathbb Q = \{\frac ab| a,b\in \mathbb Z; b\ne 0\}$. That's perfectly valid but inefficient. Notice we have include then element $\frac 34$ when we consider $\frac 34\in \mathbb Q$ as $3,4\in \mathbb Z$. But we considered it a second time when we considered $-3, -4 \in \mathbb Z$ and $\frac 34 = \frac {-3}{-4}$. And we considered it a third time when we considered $51, 68\in \mathbb Z$.
....
As to consider sets as lists with multiple listings of elements or as listings where order does matter.... well, that is why we have such concepts as multisets or sequences. Even functions is an extension of the concept.
As for a probability problem as you suggest. I imagine must would state it as something like "What is the probability that the sum of two numbers, one each drawn randomly from the collections A={1,2,2,3,3,3} and B={1,2,3,4} is at least 6?" Technically we'd say $A$ is a multiset, not a set.
Best Answer
I believe the best reference would include the reason why duplicate elements are extraneous.
It's due to the Axiom of Extentionality.
From Kunen's, Set Theory: An Introduction to Independence Proofs
Axiom of Extentionality
$$\forall x\forall y[\forall z(z\in x\iff z\in y) \iff x=y]$$
In English, two sets are equal if and only if they have the same elements.
So, $\{a,b,c\}= \{a,a,b,b,c,c\}$ by the Axiom.
On page 12, you see the explicit example that $\{x,x\} = \{x\}$.
To see why this generalizes (from a different perspective)
Notice that by the Axiom of Extentionality,
$$\forall x(z\in x\Rightarrow x\cup\{z\} =x)$$
So, adding elements to a set, that are already in it- doesn't change the set.