As a Japanese person, I'd say:
「言い草」 is often used to indicate the manner a person displays when he / she speaks. For example, if I'm a father and I ask my teenage son about school, and he replies, 「あんたに関係ないだろ」 'Why do I need to tell you?' then I might get fed up and say 「何なんだその言い草は！」 which basically means 'What way of talking is that?' and implies 'Is that the way you talk to your father?'
「決まり文句」 and 「常套句」 are really similar to each other as they are often both used to indicate a typical response. However, the difference is that, 「決まり文句」 is used to indicate a typical response made during a particular scene, and 「常套句」 is used to indicate a typical response made by a particular person. When a person always answers 「別に」 'Not much' when asked any question, and when they respond to yet another question with, 「別に」 again, I might be a bit angry and say 「それはお前の常套句だな」, but I surely won't say 「それはお前の決まり文句だな」.
「名言」 is basically a famous quote by someone. For example, 「『少年よ 大志を抱け』とは、クラーク博士の名言である。」 would be correct.
「諺」, which is 「ことわざ」 (most Japanese would prefer the latter, since the kanji is so complex), is more like a 'proverb', than just a quote. There's a similar term called 「慣用句」, but they are different. Examples of 「ことわざ」s are 「犬も歩けば棒に当たる」「果報は寝て待て」, and examples of 「慣用句」s are 「腹を立てる」「頭が切れる」. As you notice the difference, 「ことわざ」s are complete sentences or imperatives, where 「慣用句」s are incomplete sentences, or combinations of two words, giving them fixed meanings. For instance, 「頭」 means 'head' and 「切る」 means 'to cut.' However, 「頭が切れる」 doesn't mean 'the head is able to be cut', instead, it means someone 'is very smart and intelligent'.
Hope this helps!