Is the void set (∅) a proper subset of every set?

Calling $\emptyset$ and $A$ "improper" subsets of a set $A$ is not universal, and it is confusing in this case, because the meaning of "proper" is not the same. It is standard to say that $S$ is a proper subset of $A$ if (and only if) every element of $S$ is an element of $A$, but $S$ is not equal to $A$, i.e., at least one element of $A$ is not in $S$. Under this definition, $\emptyset$ is a proper subset of every nonempty set, even though it is "improper" according to the convention you were also given. Just remember that mathematical terminology varies and isn't always logical. Here "improper" does not mean "not proper". (For this reason I would personally not use the convention of calling the empty set "improper".)

When finding all proper subsets, you should count the empty set.

Given a set $A$, a proper subset is any set $B$ such that $B\subseteq A$ and $B\neq A$; that is, $B$ is contained in $A$ but is not equal to $A$. This is denoted by $B\subset A$ in some texts.

So while $A$ is a subset of itself, it is not a proper subset of itself. And this is true for any set, even the empty set (or void set, as you call it).

Speaking of which, the empty set $\emptyset$ is not only a subset of any set, but also a proper subset of any non-empty set.

Edited to include a solution to OP's new question:

You have the answer in front of you. If $A$ has cardinality $n$, then the number of subsets is $2^n$ and the number of proper subsets is $2^n-1$, because the only set we have to "throw out" is $A$ itself in order to get all the proper subsets.

So if $A=\{a,b,c\}$, then there are $2^3-1=7$ proper subsets. We can just list them:

$\{a,b\}$

$\{a,c\}$

$\{b,c\}$

$\{a\}$

$\{b\}$

$\{c\}$

$\emptyset$

A set A is a subset of a set B if every element of A is also an element of B.

A set A is a proper subset of a set B if A is a subset of B and there is at least one element of B that's not an element of A.

Thus, the void set is a subset of all sets, and it's a proper subset of every set except itself.

Also, notice that we notate the void set using $\emptyset$, not $\phi$.