Since $\emptyset \subset A$ where $A$ is any set, does that mean $\emptyset \in A$? Clearly $\emptyset \subset A$ where $A$ is any set. But does that mean $\emptyset \in A$? And if so, would it make sense to try to perform arithmetic operations with it. Like
$$\emptyset \cdot 5 \tag{where $5 \in A$}$$
This is inspired by a question that was along the lines of: if a relation is symmetric and transitive, is it reflexive? Where I've seen (and am relatively satisfied by) the answer of: no, consider the empty relation.
 A: Normally $\emptyset$ is not a number.  Multiplication is defined between two numbers.  Hence it is equally as meaningful to write $\emptyset \cdot 5$, as it is to write $banana \cdot 5$.  They mean nothing on their own, but we can always assign meaning to them.
However, there is an important exception.  In a common construction of natural numbers, due to Zermelo and Fraenkel, everything a set: there are no separate "non-set" numbers.  The number zero is defined as the empty set, the number one is defined as the set containing the empty set.  And so on, as described in the link.  In this construction, it is meaningful to write $\emptyset\cdot 5$, because this translates to just $0\cdot 5=0$.
A: First off $A \subset B$ does not mean $A \in B$.
Example:  $\mathbb Q \subset \mathbb R$.  But $\mathbb Q \not \in \mathbb R$.  If it were so, exactly which number is $\mathbb Q$ equal to?  It makes no sense.
$\subset$ compares two sets as to whether all the elements of a set or also elements of the other.  $\in$ refers to elements in a set an whether they are in a set.
It is vacuously true that $\emptyset \subset A$ for every set ($\emptyset$ has no elements; so every element it has, all zero of them, is in $A$) but it's pretty clear that it is not true that $\emptyset\in A$ for all sets $A$.  After all $\emptyset$ is not an elephant so $\emptyset \not \in \{Babar, Tantor, Haiti, pink honk-honk\}$.

And if so, would it make sense to try to perform arithmetic operations with it. Like ∅⋅5

I have to admit absolute puzzlement as to how the concept of $\emptyset \in A$ for all sets $A$ could have anything to do with defining arithmetic on $\emptyset$ so I'm not sure how to answer this.
Your explanation of empty relationships doesn't seem to make what you are asking clearer.
A: 
"Clearly $\emptyset \subset A$ where $A$ is any set. But does that mean $\emptyset \in A$?" 

If $A$ is a subset of $B$, it does not imply that $A$ is an element of $B$. 
The empty set is a subset of every set, but it is not an element of every set. 
For example, the empty set is not an element of the empty set. 
By contrast, the empty set is a subset of the empty set. 
A: Ordinary arithmetic is defined on numbers, be them integers, rationals or reals. Obviously, 
$$\emptyset\notin\mathbb Z, \mathbb Q, \mathbb R.$$
