When compressing a source sentence by removing some of its words, what are the main component besides the verb, subject and negation that one has to keep in order to preserve the grammaticality of a given sentence.
To begin with, depending on your understanding of "sentence", even in English you will find examples of well-formed utterances without verb or subject. For example, "Yes." may be a sentence. So the simple answer to your question is: no syntactic categories are essential.
More interesting are certain patterns of grammatical dependence, which vary between languages. For example, in English, if you use a finite verb, you usually need a subject, too - if you leave out the subject, the sentence is elliptical. Consider talking about the weather:
- It's raining.
There really isn't anything in particular that is raining, and yet, there needs to be a formal subject. The situation is similar in other languages, such as the other Germanic ones (such as German). Other languages, such as many of the romance languages, allow you to freely drop the subject. In Spanish, you can simply say, without a subject,
- Está lloviendo.
However, while the subject is obligatory in many constructions in many languages, it is still possible to construct subject-free sentences, such as
- Thank you!
- So what?
- Don't care.
Some of these are elliptical, some aren't. The non-ellipcital ones do not feature indicative mood, indicating that it is not so much sentences requiring subjects to be expressed, but verbs. And that is basically your answer: sentences generally lack requirements. However, once you introduce certain words in certain functions, such as a verb in the indicative, they tend to have dependencies which must be expressed for a non-elliptical construction. However, even such dependencies can usually be omitted in specific contexts.
In fact, you can remove pretty much and preserve grammaticality if you don't care about preserving meaning. The more the easier. "Go!" is a fully formed, perfectly grammatical English imperative sentence. So is "In!". The confusion comes from the well-formedness paradigm of formal linguistics and is not a real concern.
However, if you wanted to preserve some basic semblance of what the sentence was about, and you started out with a sentence with a finite verb in it, in English, you need at least the subject, the verb and whatever obligatory phrasal elements there are. So Elephants snore. or Elephants drink water.
Preserving grammaticality and preserving meaning of the sentence are two different things. If one is interested in conveying meaning and not worried about grammaticality, all function words can be removed, whereas the content words must remain. Example:
a. We are arriving tomorrow at noon by train.
This sentence can be reduced by removing all functional elements:
b. Arrive tomorrow noon train.
The functional words and elements have been removed: are, -ing, at, and by. The content words, in contrast, remain: arrive, tomorrow, noon, train. Note that the subject pronoun we can be viewed as a functional element in this case, since its meaning is certainly understood from context.
I think most native speakers of English would judge the reduced sentence to be ungrammatical, although they would understand it if they know the context in which it was produced. Such sentences are described in terms of telegraph style. Headlines are a good example of language that has had the function words removed.
While the distinction between function word and content word is not always clear, in many cases, distinguishing between the two types is not so controversial. Some widely acknowledged function words are the definite ad indefinite article, auxiliary verbs, and many prepositions. Typical content words are nouns, verbs, adjectives, and adverbs. The function words form closed classes, e.g. the number of prepositions in a language is small and limited, whereas the content words form open classes, meaning that new content words are being added to the language on an ongoing basis.