Is あごが出る = 顎が出る? Is this an idiom meaning to be exhausted?
2. 太郎は、自分の言ったことが聞いてもらえず、ふくれっ面をした。
I suppose this means "(someone) couldn't get (someone) to listen to what (someone) said; (someone) looked sullen", but I do not know who is doing what.
3. 日本の幼稚園でもたまにはあったろう。
I guess this may mean "It (bullying) probably even happened occasionally in the kindergarten in Japan", but what is the grammar of あったろう?
4. 皮肉たっぷりに言った
Should this be 皮肉ったっぷり? I am a bit puzzled by the verb 皮肉る since it seems to be formed using the "on" reading of 肉. Is this an odd case?
4. I understand from your answer that my example doesn't involve the verb 皮肉る, but if we look at that verb anyway, which WWWJDIC says is read as ひにくる, is it not an odd case to make a verb out of an "on" reading (other than the cases ~する of course)?
5. Does it mean that the table was covered in ring marks left by teacups (like below, except obviously that shows a glass of water)?
4) At Okurigana - Wikipedia, the free encyclopedia , it says "Okurigana are only used for kun'yomi (native Japanese readings), not for on'yomi (Chinese readings)". Do you think that 皮肉る (and others similar that you suggest exist) contradict that? Or is that る not true okurigana?
る is true okurigana and 皮肉る, 牛耳る, 退治る(there might be some more examples) are exceptions. These are from 皮肉を言う, 牛耳を執る, 退治する, respectively. (退治る is rarely used in these days, though.)
On a related note, could you comment on verbs like 信じる and 感じる? Is じる okurigana? Somewhere at the back of my mind is a recollection that this ~じる might be a variant of ~する, but is that correct? Are the two etymologically related?
You are correct. 信じる and 感じる are from 信ずる and 感ずる, furthermore, 信ずる and 感ずる are from 信ず and 感ず, respectively. Thus, these are a kind of -suru verbs. 演じる, 禁じる, 準じる, 命じる, 生じる, 通じる,,, belong to this group.
す and ず are the classical form of する and ずる, respectively. As you can see here, the difference between す/ず and する/ずる are basically the imperfective form (未然形 the pre -nai form) and the conclusive form (終止形), since す/ず can't be attached to the Modern negative suffix ない.