1)
わたしはたかくなりました is understandable, but わたしはせがたかくなりました is better.
3)
What you are bored at can be omitted when it's obvious from the context, but it's common to say it when つまらない is used, e.g. えいががとちゅうからつまらなくなりました. I would say うんざりしました for "became bored".
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