Problem 1:
Can any quotient $\tilde{X}$ of $X$ be embedded in $X$?
Moreover, does any (surjective) quotient map $\pi:X\to\tilde{X}$ left split with an (injective) embedding $\iota:\tilde{X}\to X$?
Problem 2:
Problem 1 vice versa!
Hint:
For the second claims we have:
A quotient map left splits with an embedding iff there is a continuous section.
An embedding right splits with a quotient map iff there is a continuous retraction
Answer:
Counterexample: $\mathbb{R}$, $\{0,1\}$
Here, since any two points on the real line can be separately separated any injective map induces as initial topology the discrete one while since singletons are not open in the real line any surjective map induces as final topology the indiscrete one.
So, we see that in here neither a quotient can be an embedding nor an embedding can be a quotient.
Moreover, since the first claims don't hold here the second ones cannot hold either.
Explanation:
This counterexample shows very good what is actually happening:
While initial topologies become quite fine final topologies become quite coarse. They might meet in between, however, in general they don't.
Outlook:
A more sohisticated counterexample is given by: $\mathbb{R}$, $\mathbb{S}$
It is a well known fact that the sphere is a quotient of the real line. On the other hand it can be shown that the sphere cannot be embedded into the real line.