How to show ambiguous context-free grammars in Chomsky normal form is Turing recognizable?
Your answer to question 1 is nice and correct.

On question 2, of course, you can use a single word $w$ to test the grammar $G$. However, it is not enough to use just one word, since the grammar could be ambiguous on another word.
Well, we can use all words to test the grammar.
Here is the high-level description of a Turing machine $R$ that recognizes $K$.

Loop through all words:

for each word $w$, try every derivation in $G$ of length $2|w|−1$.
If two derivations generate $w$ stop the machine and accept $G$.


There are many ways to loop through all words. For example, we can fix an arbitrary order of all terminals and iterate by short-lexicographic order.
Note that when $G$ is not ambiguous, $R$ will run forever. That means $R$ is not a decider for $K$. In fact, no Turing machine can decide $K$ since $K$ is undecidable.