If $G$ is a group and $N$ is a nontrivial normal subgroup, can $G/N \cong G$?

Take $G = \mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}\oplus \ldots $ and $N = \mathbb{Z}\oplus \{0\}\oplus \{0\} \ldots$

A group with this property is called nonHopfian. It used to be an open question whether there were finitely generated examples, and some examples were eventually found by Baumslag and Solitar.

The simplest one is the group ${\rm BS}(2,3)$ defined by the presentation $\langle x,y \mid y^{-1}x^2y=x^3 \rangle$, where $N$ is the normal closure of the element $r=x^{-1} y^{-1} xyx^{-1} y^{-1} xyx^{-1}$.

The relator $r$ is equivalent to the relation $x=(x^{-1} y^{-1} xy)^2$, and adding this as an extra relation gives

$\langle x,y \mid y^{-1}x^2y=x^3, x=(x^{-1} y^{-1} xy)^2 \rangle \cong$ $\langle x,y,w \mid w=x^{-1} y^{-1} xy, y^{-1}x^2y=x^3, x=w^2 \rangle \cong$ $\langle y,w \mid y^{-1}w^2y=w^3, y^{-1}w^4y=w^6 \rangle \cong G.$

You also have to prove that $r$ is not equal to the identity in $G$, which requires a bit of the theory of HNN extensions.

Yes. Let $G$ be the additive group of the complex numbers, and let $N$ be the subgroup consisting of the real numbers.

Edit in response to comment by @GA316:

$(\mathbb C,+)/\mathbb R$ is clearly isomorphic to $(\mathbb R, +)$, and it is well known (but this requires the Axiom of Choice) that $(\mathbb C,+)\cong(\mathbb R,+)$.

My answer is a special case of the answer posted simultaneously by Asaf Karagila, considering $\mathbb C$ as a vector space over the field of rational numbers.