The answer to the first question is "yes". See this paper of the Dokchitser brothers, Lemma 3.1 for the case where $K/\mathbb{Q}_p$ is Galois. In the general case, apply the result to the Galois closure $K'$ of $K$ to get $F'$, identify the Galois group of the local fields with a decomposition group $D$ at $p$ inside the global Galois group and take the fixed subfield of the subgroup of $D$ corresponding to $K$.
As Kevin says, unless I am missing something, the Dokchitsers' proof works with minor modifications for all three of your questions. Note that the result for question 3 follows from the previous two (using the slightly more general version of qn 1 in the above link): first take an extension in which $p$ is totally split, then work with each of the places above $p$ separately, using the answer to qn 1.