You have a wonderful grasp of Chinese grammar! It seems to me that most of your questions relate to the rhythm of Chinese speak patterns rather than grammatical necessities. (1) Unless absolutely necessary (and no counter example comes immediately to mind), the Chinese language abhors ending a phrase with a verb that is monosyllabic. All Chinese speakers will understand 互相幫, but somehow it sounds incomplete. That is why Chinese is full of compound verbs and nouns that are repetitive - like 幫助、游泳、牙齒、房屋、書本、衣服…There are two ways to end a phrase without using a monosyllabic verb - either by using a direct object (noun) or using a compound verb, adding a particle or an adverb. For example, it sounds "bare" to say 拿！On the other hand, 拿好，拿穩，拿著 all sound complete. Even 我們安靜聽 sounds a little incomplete, so Chinese speakers will somehow try to round it out by adding another syllable, like 我們安靜聆聽。As for the second part of the sentence, it is exactly because of the aspect particle that you can shorten the verb, because with 著 the verbal phrase (not including "to think" - that is a separate verb) has two syllables. (2) Actually, both 幫and 助 independently mean "to help." As noted in (1) above, modern Chinese has a rhythmical preference for verbal phrases longer than one syllable, especially to end a sentence or phrase. (3) Yes, great translation. (4) You do not strictly need 著 in the second part of the sentence if you have something else in its place. In fact, you can actually say 遇到難題時我們互相幫忙想辦法. I think here, you simply cannot say 幫想辦法 because 幫 and 想 are separate verbs and Chinese does not like juxtaposing two independent monosyllabic verbs. In sum, it seems to me that most of your questions relate to the rhythmic flow of the language than to grammatical requirements. I hope this was helpful.

1.There are lots of such compound verbs in Chinese in which the two single characters mean the same thing. As for 帮助， 帮=助（in meaning)，1)when used as a vt. 帮助+object=助+object=帮+object 2)when used as a vi. we usually use 帮助 or 帮， but we don't use 助 alone。 when used together with互相，we use 帮助ratehr than 帮，(我们应该互相帮助） when we use 帮alone，the sentence is always as simple as"你帮不帮?/我帮/我不帮/我不想帮" ，and in all these sentences, we don't use 帮助， it is more like a habit. 3）as for 帮着, here 着 indicate that a state continues. We see "verb+着" as a whole, we'd better not take them apart. And another thing i want to point out is that when combined with other characters to make a new verb or phrases, if the two characters mean the same thing, we'd better just use one of them: 帮着。 2.帮助 doesn't mean the result of an action. as you can see , the structure of the former words are different from that of 帮助 。 见 is the result of 看， 到 and 住 are not verbs in the compound verbs, they are more like adverbs that mean a result。 and in 帮助， 帮and 助 mean the same thing, and they are both verb。 3.When faced with trouble, we should help each other out/we should help each other to figure out ways. 4.着　here means the state which is ongoing, but not the action which is ongoing.

1 帮，助are same meaning, they both mean help. So 帮助just a formal saying of help in Chinese, and you will find,there are lot of such two character words mean same action and used as a whole verb, like 居住（both mean live in），选择(both mean choose). 帮助着is no problem in grammar, but we just don't say like that. There is one rule that when there is only one verb in sentence, most of time we use the two character one(here not just for formal reason), that in fact just for the rhythm of the sentence, Chinese is alway a beautiful language. And for the difference, 帮助 can also be a noun, ex. 这对我很有帮助。That is a great help to me. 帮just a verb, can't be noun. 2 then the second question has already answered. 3 you are rigth 4 着also mean the verb before it is happen same time with the later one, as a way of the later one acting. Here, the later one is 想, so how to think? help to think. Another ex. 我躺着看书。when I read book, I lay on bed. It should be "I read laying on the bed"? This also not a ongoing thing, but a kind of usually way I doing things. Of course, ongoing things is also one use of 着。

1) because 帮助着 isn't 好听 Chinese tend to shorten rather than extend phrases 2) no, 帮助 isn't doesn't have any result-ative meaning 3) yes 4) same reason you need Continuous (Progressive) tenses in English in 3) you give "rough translation" and then according to it you are trying to make out a Chinese language rule don't confuse Chinese vocabulary with English grammar