Exercise about First-order logic
It might be useful to take a look at the ZFC axioms. The existence of one element sets is guaranteed by the Axiom of pairing.

The axiom of pairing says that given $A$,$B$ sets, one can construct a third set $C = \{A, B\}$. Formally:
$$ \forall A. \forall B. \exists C.\forall D. (D \in C \iff D = A ~\lor~ D = B) $$

A singleton set is just the pairing $\{A,A\} = \{A\}$.

The axiom of union says that for any set $A$ there is a set $U$ whose elements are exactly the elements of the elements of $A$. That is:

$$ \forall A. \exists U. \forall c. (c \in U \implies \exists D.(c \in D ~ \land ~ D \in A)) $$


  A set $X$ is the union of all its subsets containing just one element.


Let's look at some example, let $X = \{a, b\}^\star$ be a set, take $X^\prime = \{\{a\},\{b\}\} $, which is also a set by the axiom of pairing. By the axiom of union there exists a set $\cup X^\prime$ such that:

$$ \cup X^\prime = \{a,b\} = X$$

So you can look at the Union Axiom and try to modify to fit your problem, you have to make a union on a set of singleton sets, which means you have to find a way to define $X^\prime$. 

$\star$: I'm assuming $a$ and $b$ are also sets.




  EDIT: Regarding problem 4.: My main problem is that I am not really
  sure, how to express or write down the solution - I am not too sure,
  what is allowed, what not. Thus I take any idea that seems remotely
  right or constructive.


Well, from your previous answers it seems you're only allowed to use basic operations like membership and equality, if more stuff is allowed you can literally just write something that looks like natural language while still being formal.