Your question concerns the extended meanings of compound complement of direction (复合趋向补语的引申用法)in Chinese Grammar.In Chinese,some of compound complements of direction have extended meanings just like 下来&下去 in your examples. 1.下来：used after the verb predicate expressing the gradual change from the past to the present,from the start to the end. 2.下去：used after the verb predicate indicating the continuation of an action. So in your examples "坚持下来" means somebody has tryed doing sth.from the past to the present,from the start to the end. . Whereas "坚持下去" means somebody is persevering in doing sth. now, and will continue to the end.

1.in terms of meaning: “记得”means "bring someone or something out of one's mind" “记住”means "remain something or someone in one's mind and insure it is not forgotten" sometimes they can replace with each other. sometimes can not.Try to feel the distinction by examples e.g.我记得他的名字。(I remember his name.)or 我记住了他的名字(I kept his name in my mind.) e.g.你要记住（记得）走得时候关窗。(both ok) e.g.老师让我们记住这个公式。（can't use 记得）(The teacher asked us to put this formula) e.g.我记得你有个姐姐。（can't use 记住）(I remembered that you have a sister.) 2.in terms of grammar: both 记得 and 记住 can be followed by a noun,pronoun,or another verb-phrase.And 记得 can be followed by a accusatival clause.whereas 记住 can not. e.g.我记得你以前去过北京。（记住 can not be used） e.g.他记得睡觉前要打电话给女朋友。（记住 can not be used）

1.in terms of meaning. 1）认为&觉得 both express judgement of something or someone,and sometimes they can replace with each other. 认为 emphasize particularly on showing the opinion after reflect and consideration,the opinion is often based on analyse,comprehension and cognition,so the tone for 认为 is definite , While the tone for 觉得 is not very definite, as 觉得 is just by right of individual feeling. 2）And another meaning of 觉得 is "come into being some feeling"，whereas 认为 doesn't have.. e.g.我觉得累。 e.g.我觉得很忙。 e.g.我觉得很伤心。 e.g.我觉得特别难受。 2.in terms of grammar 认为 can be often used in "被--字句"（passive voice）, 觉得 is rarely used in"被--字句"(passive voice). e.g.他被老师们认为是班上最听话的学生。 e.g.这个电视剧一致被认为是近年来的经典之作。

1.一下 has two usages in Chinese： 1) as an adverbial usually before the verb means "all at once" or "suddenly". e.g.刚才，他把坏消息一下全告诉我了。(He told me the bad all at once just now.) e.g.我一下想不起他的名字了。(Suddenly, I fogot his name ) e.g.他一下吃完了桌上的馅饼。（He ate up the pies on the table all at once.） 2) as complements after verbs. to describe the frequency of an action. Just like "once "in english. e.g.那条路出事故了，有两辆车撞了一下。(There is an accident in that road.Two cars collided each other once.) e.g.他肯定睡着了，我拍了他一下，他都没反应。(He must be asleep.It doesn't work for him even though I patted him once.) In terms of Grammar "一下" is a Numeral-Measure Word phrase. In Chinese, measure words can not only embellish a noun.but also a verb. This kind of phrase embellishing a verb is called Verbal Numeral-Measure Word phrase. 2.“一下”is often used to imply that the action short or quik and to express a sense of being light and relaxed or a meaning in a circumlocutory tone. e.g.你看一下，这是你的钱包吗？(Have a look ,is this your wallet?) e.g.你能帮我一下吗?(Could you help me?) e.g.太累了，让我休息一下。(I'm so tired,let me have a rest,please.)

类似这种意思的成语有很多，不能从直译去理解，要从引申的意思去体会： 异曲同工， 异名同实， 异口同声 殊途同归， 异路同归， 异途同归

1.I'm sure the answer is 我没在吃饭。 In Chinese grammar,"zài" + verb" is one of patterns indicating that an action is in progress.And the negative form of ,"zài" + verb" is made by placing "没" before the predicate verb.The pattern should be : "没+zài + verb" e.g.我没在看电视(I'm not watching TV) e.g.他刚才没在听我说话。(He was not listening to me just now.) note that,the progressive of an action may take place either in the past ,the present or the future.if the time is not clearly indicated or shown by the context,we may generally think the action takes place in the present. 2.FAQ about不 （bu）and 没(mei) Last week one of my students in class told me that her boyfriend wanted to buy an iphone4 last Sunday,but he hasn’t bought it because it’s too expensive.She said in Chinese“太贵了(tài guì le)，他不买(tā bù mǎi)”, and then I told her that in Chinese it should be “太贵了(tài guì le)，他没买(tā méi mǎi)”.So when can we speak不(bù) and when can we use 没(méi)? As most foreigners have known, 不 （bu）for 不是 (bu shi) or for 不（bu）+verb，没（mei）just only for 没有(meiyou). Actually,不 （bu）and 没(mei) are both used to negate, but they are used differently when we want to express some negative state of an action. Here are some difference between 不 （bu）and 没(mei). 不(bù) is used to negate actions at the moment or in the future.But 没(méi) is used to negate actions in the past.Please try to feel the distinction the following sentences. e.g.我(wǒ)不(bù)回(huí)家(jiā)。(I’m not going home.) 我(wǒ)没(méi)回(huí)家(jiā)。(I haven’t gone home. ) e.g.他(tā)不(bù)出(chū)差(chāi)。(He isn’t going for a business trip.) 他(tā)没(méi)出(chū)差(chāi)。(He hasn’t gone for a business trip.) e.g.他(tā)不(bù)去(qù)吃(chī)午(wǔ)饭(fàn)。(He isn’t going to have his lunch) 他(tā)没(méi)去(qù)吃(chī)午(wǔ)饭(fàn)。(He hasn’t gone to have his lunch.) Then I think it would be prefect to express themselves exactly in Chinese if foreigners can master this point.

1.在 added before the predicate(verb) is used to indicate that an action is going on or in progress. e.g.我在开会。 e.g.他在睡觉。 e.g.她们在看电视。 (the action of 开会/睡觉/看电视 is going on) 2.着 occurring after the predicate(verb) express the continuation either of an action or of a state as the result of an action. e.g.他飞快地跑着.(the continuation of the action 跑) e.g窗户开着，门关着。(The action of 开and 关,have been finished, but the states of 开and 关 continue.) e.g.他穿着一件蓝T恤。(The action of 穿 has been completed ,but the state of the result of 穿continues.) 3.In Chinese,"verb+着"is often used before the predicate(another verb) to illustrate the means or the manners of the latter action. e.g.他坐着听音乐 e.g.他看着报纸吃早饭

1.in terms of meaning: 不能 means "can not" or "is not able to " e.g.我不能和你一起去北京。 e.g.这是他的秘密，我不能告诉你。 不行has two meaning.one is "can not or to be allowed" the other " be good at". e.g.明天和你去北京？我不行。(Go to Beijing with you tomorrow? No,I can't.) e.g.你想请假？不行! (you want to leave? No, you're not allowed) e.g.你觉得我中文不错？我不行,我只会一点儿。(You think my Chinese is not bad? No,I am not good at Chinese,I just know a bit. ) 2.in terms of Chinese structure 不能 as an auxiliary verb. have to be followed by verb and phrase of verb.But 不行 can not. and it is used alone. e.g.A:下个星期，你能陪我去看电影吗？ B:不行，下个星期我很忙。（or 下星期我很忙，不能陪你去看电影） 3.a set phrase of 不行 is 不行了。 不行了 means can not do something anymore. e.g.我不行了，我吃得太多了。（I have eaten too much,I can't eat anymore.） e.g.太累了，我不行了。(I'm tired very much,I can't do it anymore)