How to answer the following queries on a tree?
Without any precomputation, a depth-limited version of depth-first search would be a natural approach to this problem: then it's like finding the minimum of an array, you could find the minimum weight during the search. But a time-efficient way (if you have multiple queries for different nodes and $K$) is to indeed precompute the answer for all the nodes: a linear algorithm seems achievable, starting from the bottom level up to the root, level by level. You have to store for each node an array $a$ such as $a(K)$ gives the desired value. And if you modify the value of a node later, I think a backpropagation scheme could be also considered.