Is $\emptyset$ a set element? Is there a difference between $\{1,\emptyset\}$ and $\{1\}$? Does the former have $2$ elements and the latter $1$ element? Does adding a string of $\emptyset$'s increase the number of elements of a set?
 A: To your first question - Yes, the two sets are different. In fact $\{1\}\subset\{1,\emptyset\}$.
Adding a string of anything doesn't increase elements in a set - $\{1,2,3,4\}=\{1,1,2,3,3,3,3,4,4\},$ so no this doesn't change anything for the empty set either.
A: The main problem here is that $1\in\mathbb N$ and $\varnothing$ are different in nature.
$1$ is an integer, while $\varnothing$ is a set.
You can't have a set mixing sets and non-sets, $\{1,\varnothing\}$ is simply wrong in general. 
Remark : emphasis on "in general", because as in Unix we say "everything is file", for set theoretists "everything is set", naturals, reals, functions, etc... See below for instance the construction of naturals, for a possible meaning of $\{\varnothing,1\}=2$.

$\{\{1\},\varnothing\}$ would be fine and it has $2$ elements which are sets.
The sets $\{\varnothing\}$ and $\{\varnothing,\varnothing\}$ are equal, in the same way that $\{1\}=\{1,1\}$ because in a set, elements appear only once.
So repeating an element won't increase the cardinality, it's just the same set.

Now, the fact that for a set $S$ we have $S\neq\{S\}$ is the base of the definition of the axioms of the naturals. There are two main constructions :


*

*Zermelo 


$\begin{array}{l}
0=\varnothing\\
1=\{0\}=\{\varnothing\}\\
2=\{1\}=\{\{\varnothing\}\}\\
3=\{2\}=\{\{\{\varnothing\}\}\}\\
n+1=\{n\}=\{\{..\{\varnothing\}..\}\}\}
\end{array}$


*

*Von Neumann


$\begin{array}{l}
0=\varnothing\\
1=0\cup\{0\}=\{0\}=\{\varnothing\}\\
2=1\cup\{1\}=\{0,1\}=\{\varnothing,\{\varnothing\}\}\\
3=2\cup\{2\}=\{0,1,2\}=\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}\}\\
n+1=n\cup\{n\}=\{0,1,2,..,n\}=\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\},\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}\},...\}\\
\end{array}$
In the Zermelo case, the cardinal of a number is always $1$ but elements are  themselves sets of cardinal $1$ nested like russian dolls.
In the Von Neumann case, the cardinal of the number, is the number itself, but the nesting is more complex. Every natural is the set of all its predecessors.
A: Yes they are different sets. The empty set is not nothing. It is the set which contains nothing, an empty box, if you will. Thus the set which contains both one and an empty box is different than the set which contains just one. Incidentally, the first set you mentioned is what we define the ordinal 2 to be. 
