The cross product of two sets

Yes. $A \times B$ is a set of ordered pairs of elements from $A$ and $B$. If $x \in A$ and $x \in B$, then the ordered pair generated will be $(x,x)$ which is a valid ordered pair and is an element of $A \times B$.

For instance, if $A = \{0,1\}$ and $B = \{1,2\}$, then $A \times B = \{(0,1),(0,2),(1,1),(1,2)\}$. Hope this clarifies.

Yes, $A\times B$ is the (new) set of all ordered pairs $(x,y)$ with $x\in A$ and $y\in B$. There will be no problem if $A$ and $B$ share common elements since an ordered pair is precisely such that it can contain duplicates (an unordered pair, i.e. a set $\{a,b\}$ can not). So, for example, $A\times A$ is just the set of all ordered pairs of elements of $A$. For example, $\{1,2\}\times\{1,2\}=\{(1,1),(1,2),(2,1),(2,2)\}$.

EDIT: Ordered means that the order of the elements matters, that is, $(1,2)\ne(2,1)$ whereas $\{1,2\}=\{2,1\}$.