In either in writing or speech, can we contract plural noun with the present perfect? For example, can we say or write:
The children've eaten.
Yes, absolutely. You run the risk in writing of seeming informal, so it isn't ideal for academic work, etc., but if you're looking to strike a tone of brisk-but-earnest, it works well in speaking. Then again, it's a dialectical variation in most forms of English: it's incredibly common in the Southeast, for example, but maybe not in other areas.
I do believe that your sentence above is grammatically correct, however using it will make writing sound more informal. Even using the contracted form of 'have' with a singular word such as 'child' will immediately make the words sound less formal. For example:
The child's eaten.
As opposed to the more formal:
The child has eaten.
This might be due in part to the fact that usage of contractions must have developed after people began to use common word patterns in everyday speech. Generally, in my experience, people will speak in a more informal manner than they write.
To a lesser extent, if you wish the writing to preserve a degree of formality but are looking for a way to put a contraction in then it might be slightly better to use a pronoun in place of the noun:
They've gone for a walk.
The same rules apply for plural nouns. If you wish for the style of writing to be formal (e.g. a job application letter) then it is better to avoid contractions as a rule of thumb.
Yes, we may employ this contraction. As stated above it is a matter of style in some cases, as well as the response you desire from the audience/listener. It seems to me that by not employing this contraction, when speaking in contemporary America, one might sound a a bit formal, even imperious. If that is your desired effect, then using the more formal "the children have" will accomplish your goal. I've heard people eliminating the contraction as a subtle, and distasteful, attempt at mocking a less educated conversationalist.
Yes, especially if you're quoting spoken word. Even the queen says'gonna'. In formal use, some contractions (like the infamous 'it's') are more common in writing than others, such as the one you have mentioned. However, do check your usage for ambiguity before you commit, especially when the contraction is 's'. There is a lovely example quoted above: "The child's eaten". What about the vegetables? Did they eat them as well? Sorry, couldn't resist . .
Regarding the example regarding the informality of a contracted form of have with a singular noun, how would you know if "The child's eaten" is present perfect or passive? The child has eaten or The child is eaten.
have
has been contracted to've
separately from the preceding word. I'd write a space between the two -- "The children 've eaten."