Sorry, the first version of this answer was broken in a few ways.

For your first question, it seems that there is more than one construction that specializes to what you may choose want. For example, you can take the completion $\hat{X}$ of a prime ideal variety $P$, localize to get X$ along a closed subvariety $Z$, and then take the local ring at tensor product $P$, \mathscr{O}_{\hat{X}} \otimes_{\mathscr{O}_X} K_X$, where $K_X$ is the function field. Alternatively, if you have an effective divisor $D$ in a variety $X$, you can take the scheme whose underlying topological space is $P$-adic completionD$, and whose sheaf of rings is given by $U \cap D \mapsto \varinjlim_m \varprojlim_n \Gamma(U, \mathscr{O}_X(mD)/\mathscr{O}_X(-nD))$. These two constructions are identical when we are presented with a codimension one subvariety, then localize againsuch as a point in a line. I don't do not know if there is a special succinct name for this composition of operationseither construction.

For your second question, you can ask for the modules to be have a $k$-linearly topologizedk$-linear topology, such i.e., there is a basis of neighborhoods of zero formed by $k$-submodules. We can then demand that the action of $k((x))$ is continuous, where $k((x))$ is given the $x$-adic "usual" topologyand in particular , with $k$ is discrete . Then and $ \{ x^n k[[t]] \}_{n \in \mathbb{Z} } $ forms a basis of neighborhoods of zero. A continuous $k((x))$-module is the same as a $k$-linearly topologized topological $k$-module equipped with a continuous invertible topologically nilpotent endomorphism. One possible reference for generalities is EGA1 0.7.

For your first question, you may choose a prime ideal $P$, localize to get the local ring at $P$, take the $P$-adic completion, then localize again. I don't know if there is a special name for this composition of operations.

For your second question, you can ask for the modules to be $k$-linearly topologized, such that the action of $k((x))$ is continuous, where $k((x))$ is given the $x$-adic topology and in particular $k$ is discrete. Then a continuous $k((x))$-module is the same as a $k$-linearly topologized $k$-module equipped with a continuous invertible topologically nilpotent endomorphism. One possible reference for generalities is EGA1 0.7.