Finding missing edge weights in the context of minimum spanning tree
Firstly, to make the nice answer given by OP to Problem 1 the only valid answer, we should change its question &quot;what are $x$ and $z$?&quot; to &quot;what are the maximal weights of $x$ and $z$?&quot;. Otherwise, we can have many valid answers such as &quot;$x=3, z = 5&quot;.
Secondly, to make the nice answer given by OP to Problem 2 the only valid answer, we should change its question &quot;What are the weights of other edges?&quot; to &quot;What are the minimal weights of other edges?&quot;. Otherwise, we can have many valid answers such as &quot;$w(ED)=16, w(CD)=115$ and $w(AB)=19$&quot;.
Lastly and most importantly, let us answer the central question, &quot;Is there some standard well defined steps for both problems? Or its more sort of trial and error work?&quot;. A standard way to rephrase the question is, &quot;Is there an effective algorithm for both kinds of problems?&quot;
Let $G$ be an edge-weighted connected undirected graph with some unknown weights. Let us state the questions clearly in more detail. We will assume there are graphs that satisfy the given conditions in the problems. (Otherwise, we can add extra steps to the algorithms below to detect those impossible situations).
The first kind of problem: unknown weights in an MST

Assume that we know a minimum spanning tree (MST) of $G$ such that all edges of unknown weights belong to that MST. What are the maximal values of those unknown weights?

Here is the sketch of an effective algorithm, which can be considered as a precise abstraction of OP's concrete solving procedure.
Pick any edge $e$ that is not that MST. If we add $e$ to that MST, we will create a unique simple cycle. All edges of MST in this cycle must be as light as or lighter than $e$. Collect these inequalities, all of which have the form &quot;$x\leq a$&quot; where $x$ is the unknown weight of some edge in that MST and $a$ is a known number. Repeat until we have gone through all edges that are not in that MST.
Now we have a collection of inequalities about some unknowns and numbers. Remove redundant inequalities. Find an inequality &quot;$x\leq a$&quot; with the smallest number $a$ where $x$ is an unknown. Let $x=a$  be part of the answer. Repeat this process until we have no more inequalities.
The second kind of problem: unknown weights outside an MST

Assume that we know a minimum spanning tree (MST) of $G$ such that all weights of edges of the MST are known. What are the minimal values of the  unknown weights?

Here is the sketch of an effective algorithm, which can be considered as a precise abstraction of OP's concrete solving procedure.
Pick any edge $e$ of unknown weight. If we add $e$ to that MST, we will create a unique simple cycle. All edges of MST in this cycle must be as light as or lighter than $e$. Collect these inequalities, all of which have the form &quot;$a\leq x$&quot; where $x$ is the unknown weight of $e$ and $a$ is a known number. Repeat until we have gone through all such edge $e$ of unknown weight.
Now we have a collection of inequalities about some unknowns and numbers. Remove redundant inequalities. Find an inequality &quot;$a\leq x$&quot; with the largest number $a$ where $x$ is an unknown. Let $x=a$ be part of the answer. Repeat this process until we have no more inequalities.
More general problems?
If you have read this far, it will be obvious that we could create more general problems of similar kinds. We will have a general approach towards producing an algorithm as well. So much for my answer that is already long.