Let $k$ be a number field with ring of integers $\mathcal{O}_k$, let $|\cdot|_v$ be the non-archimedean absolute value associated to a prime ideal $\mathcal{p}\subset\mathcal{O}_k$, and let $k_v$ be the completion of $(k,|\cdot|_v)$. Then $k_v$ contains the complete local ring $\mathcal{O}_v$ and its maximal ideal $\mathfrak{p}_v=\varpi\mathcal{O}_v$, where $\varpi$ is a chosen uniformizer. As any element of $\{u\varpi:u\in\mathcal{O}_k^\times\}$ is also a uniformizer (none of which is canonical, as far as I know), I am curious if there is a choice of uniformizer that is preferable in some way.
This question occurred to me because we usually write $p\mathbb{Z}_p$ for the maximal ideal of $\mathbb{Z}_p$, seemingly favoring $p$ as a uniformizer over any other $up$. I think this is sensible though, as $p\in\mathbb{Z}$. In the more general setting above, a favorite choice is less clear to me. We may choose $\varpi\in\mathcal{O}_k$ if desired but it is not generally possible to choose $\varpi\in\mathbb{Z}$. With the nice properties of $p$ in mind, my questions are:
Do you know a reason to choose a particular uniformizer over the other possible choices?
For which $k_v$ is it possible to choose a uniformizer $\varpi\in\mathbb{Z}$?