untmdsprt wrote:振る(ふる) , 断る(ことわる) - these could be a simple case of intransitive and transitive verbs, but if it's not then what's the difference?
Both are transitive in this case, but generally transitive/intransitive pairs use the same kanji with different okurigana, e.g. 壊す/壊れる, 冷やす/冷える. 振る and 断る are two separate words.
振る has a ton of meanings, but I think the one you're after is 'to reject, turn down, dump'. It seems to be used mostly in the passive when somebody gets dumped by their partner or rejected during ナンパ or 告白. 断る means, more generally, to turn down or reject a request in all sorts of situations.
untmdsprt wrote:答え and 返事
These can be interchangeable, but you most often hear 返事 used to refer to a reply to an e-mail, text, letter etc, and 答え as the answer to a question.
If I have this problem, I find that J-J definitions are useful. Since the problem is in using the words though, I think it's best to try and see them being used as much as possible. Ideally this would happen naturally through reading, conversation, watching TV etc, but if you need to force the issue, try typing the word into Google or using something like Jim Breem's example sentence search.