Empty set clarification Is $∅ ∈ \{\{∅\}\}$ true or false?
I think it is false and my reasoning is this,

∅ is an element of a set of subset ∅

Since ∅ is an element of the set, it is therefore not an element of the subset inside the set. Am I right?
Is ∅ ⊄ {∅,1,2} true or false?
I think it is true and my reasoning is this,

∅ is not a subset of set ∅, 1, 2

∅ is an empty set therefore it is a subset of {∅, 1, 2}
Lastly, is it right to say that in any power set of X, ∅ will always either be ∈ or ⊆ of power set X?
 A: The emptyset, which we'll denote $\varnothing$, is a subset of every set, and it is even a subset of the empty set itself.
$\varnothing \subset \{\{\varnothing\}\}$ but $\varnothing \notin \{\{\varnothing\}\}$.  Our set has one single element, $\{\varnothing\}$, the set containing the emptyset, so $\varnothing$ is not an element of the initial set.
However $\{\varnothing\} \in \{\{\varnothing\}\}$.
And $\{\{\varnothing\}\} \subseteq \{\{\varnothing\}\}$.

In your second example, let's call $S= \{\varnothing, 1, 2\}$.
Then $\varnothing \subseteq S$, because the emptyset is a subset of every set.   $\varnothing \in S$, as well, as it is an element of S. And $\{\varnothing\} \subseteq S$, just like $\{1, 2\} \subseteq S$.
So the powerset of S, the set containing all subsets of S is as follows:
$$\{\varnothing, \{\varnothing\}, \{\varnothing, 1,\}, \{\varnothing, 2\}, \{1\}, \{2\}, \{1, 2\}, \{\varnothing, 1, 2\}\}$$
A: 
Is ∅ ∈ {{∅}} true or false?
  I think it is false and my reasoning is this,
∅ is an element of a set of subset ∅
Since ∅ is an element of the set, it is therefore not an element of the subset inside the set. Am I right?

You put that a little awkwardly, but as I understand it, your reasoning is: "I see that $\emptyset$ is an element of $\{ \emptyset \}$, but that $\{ \emptyset \}$ is itself an element of the whole set $\{ \{ \emptyset \} \}$ (as you put it: the $\{ \emptyset \}$ is 'inside' the $\{ \{ \emptyset \} \}$), and it is therefore not an element of that 'larger' set $\{ \{ \emptyset \} \}$.
OK, that doesn't follow. Just because $X \in Y$ and $Y \in Z$ does not mean that $X \not \in Z$. For example, think of the set $\{ \{ \emptyset \}, \emptyset \}$.
A: Your conclusion is correct, but the reason you give is wrong. 
Your reason is : since $\emptyset$ it is a subset ( a member of the power set) of the set under consideration,  it is not an element. But consider this counter example, where 0 is the number zero : $\{ 0 , \{0\}\}$ . The set $\{0\}$ is a subset , but also an element. 



*

*The reason you could have given is : the fact that $\emptyset$ is an element of an element of the set under consideration does not imply that it is also an element of the set under consideration; witness, the empty set can be seen nowhere " on the list" of the elements of the set under consideration. 

*Determining whether something is an element of a set is as simple as this : make a list of all the elements; if you cannot find the object on the list, it means it is not an element of the set. 
The list of the elements of $\{\{\emptyset\}\}$ is very short. 
It looks as this. List : $\{\emptyset\}$. 
So 1 and only 1 element, which is a singleton set. 
Do you see $\emptyset$ on this list? No, on the list you see a set that has one element, but the empty set has no element.

