it seems you'd better study a bit rules of Chinese words' creation, it maybe as hard as how English words came from Latin. I'll try to explain from what I can tell, also please try to correct my English. ;) "忙" is the root of these words, which means more closely to "busy", less close to "hurry". The left part of this character means "heart" or "mind", right part means pronunciation. Then here are the four words: 匆忙，急忙，赶忙，连忙, they are all adverbs & talking about how busy or hurry the action could be. The 匆匆忙忙 & 急急忙忙 can be words is because "匆匆", "急急", "忙忙" are significant words; while "赶赶" is not a word, "连连" is a word but "连连忙忙" would be confusing from the meaning. "匆匆" used to be a word to describe the time goes too fast, hence "匆忙" has the meaning of "to be busy to save time", but it's not too rush. It mostly closes to walking action, eg: 匆忙赶路 hurrying on one's way, 匆忙离开 to leave in a hurry. Normally we don't use "匆忙" to express a transition between 2 actions. BTW, "匆匆忙忙" has the similar meaning but feels more emotional & a bit slower from "匆忙". "急急" has the meaning of "to consciously rush" because it has a "心" inside the character. So "急忙" has the meaning of "to consciously rush on". We normally use "急忙" to express a transition between 2 actions. "急急忙忙" also has the similar meaning but feels more emotional & a bit slower from "急忙". "赶忙" also has the meaning of "rush". However the "赶" has a "走" inside the character, so it has more meaning about the real action. It's less using to express a transition between 2 actions than "急忙". "连忙" means at once promptly. “连” means "to link". So this one is shorter, and mostly has to be a transition between 2 actions. But then, we don't say "连连忙忙" because it can't be extended; whereas "匆匆忙忙 & 急急忙忙" don't have meaning of transition between actions.

匆忙 you’re walking fast. It focuses on your behavior. 我走得很匆忙；他形色匆忙；我走得太匆忙以至于忘了拿钥匙 急忙 it’s meaning is the same as “赶紧”，you’re in hurry, it focuses on your mentality. 我急忙跑出去追她 赶忙 it is not a favorite word 连忙 it focuses on you want say something or do something in hurry. 我连忙道歉;我连忙把它捡起来

连忙是副词（adv） 表示人的行为动作迅速或者急迫，赶快，急忙。一般用在陈述句和描写句中。eg:1，看到客人已经坐下，她（连忙）端上茶来！ —— 强调动作的迅速和连贯性 she have already prepared! she did it very quickly! 2 ，看到客人已经坐下，她（急忙）端上茶来！——强调动作做得急急忙忙，she have not already prepared! she did it in a hurry. 3,她赶忙/赶紧/连忙端上茶来！——‘赶忙’usually used in oral chinese

first, u should know in chinese there are 4 kind of words can 重叠, verb,adjective,noun and quantifiers. sometimes 匆忙and 急忙are adjectives and sometimes they can be advers too,so when they are adjectives can 重叠.but 连忙and 赶忙are adverbs just,so they can't 重叠. when 匆忙and急忙 are adjectives,they are diffrence from 连忙and赶忙.匆忙和急忙except can be used to describe people's hurry action and also to people's emotion.eg:你怎么这么匆匆忙忙（or 急急忙忙）的，时间还早呢. but for 连忙和赶忙，we can't put them in this scentence,they just be used to bedeck verb.eg:趁明天之前连忙（赶忙）把作业做完. hope these can help u.

连忙<匆忙=赶忙<急忙 This is just my own idea.