Q1: Do you known examples, where the different is not a principal ideal?
Q2: Is there a good interpretation for the reason, why this happens?
See e.g. Neukirch, Proposition 2.4, page 197.
The reason why I ask: the definition of the canonical additive character $x \psi:x \mapsto \mathrm{e}^{2 \pi i (\mathrm{Tr}_{F / \mathbb{Q}} x \mod \mathbb{Z})}$ is somewhat unsatisfactory, if I want to identify the Pontryagin dual of the additive group of $\mathfrak{o}$ with the additive group of $F/\mathfrak{o}$, which simplifies some of notation involved when computing some $p$ adic integrals or Gauss sums.
Btw. with the canonical additive character, the Pontryagin duality is of the following form:
$$F / \mathfrak{D}^{-1} \cong \mathrm{Hom}_{ab.group} ( \mathfrak{o} , \mathbb{C}^\times).$$ where $\mathfrak{D}$ is the different and the isomorphism is given by $$ \xi \in F \mapsto \psi( \xi \cdotp).$$
