I'm trying to find out what is the grammatical category that corresponds to such expressions that use to introduce clauses, such as:
Any technical explanation of any name for this kind of expressions will be appreciated.
It would also be helpful to know what is the name of the kind of sentences that include these expressions.
Traditionally, linguists do not consider these forms to be a single constituent in English. Rather, it is thought that the entire that-clause is a subordinate clause embedded in another clause, called the matrix clause, as an argument of the matrix clause. For instance, if we have the sentence It is not the case that the sun rises from the west, then that the sun rises from the west is a complement clause of not the case, which would be called a 'complement-taking predicate' in Noonan's terminology. A classical treatment of these clauses is Noonan (1985/2007).
However, Thompson (2002) presents an alternative view that is more in line with what you have in mind. She argues against the traditional analysis, and calls these epistemic/evidential/evaluative fragments. Under her analysis, all three of your expressions would be known as epistemic fragments. It is nice that and It sucks that would be examples of evaluative fragments, and [subject] heard that and It's said that would be examples of evidential fragments.
Noonan, Michael. 2007. Complementation. In Timothy Shopen (ed.), Complex Constructions (Language Typology and Syntactic Description), vol. 2, 52–160. Cambridge: Cambridge University Press.
Thompson, Sandra A. 2002. “Object complements” and conversation: towards a realistic account. Studies in Language 26(1). 125–163.
They are not constituents, but just parts of ones that are best called fragments.
In full, as in for example "It is highly unlikely that Ed will turn up", they are extraposition constructions, in which the dummy pronoun "it" is subject and the content ("that") clause is in extraposed position, outside the VP.
That Ed will turn up is highly unlikely. [basic version]
It is highly unlikely that Ed will turn up. [version with extraposition]
Note that in the extraposed version, the predicative complement is just "highly unlikely", just as it is in the basic version.
There is some syntactic evidence first noticed by Jerry Morgan that certain apparent topmost clauses are more like qualifying adverbs. That involves the agreement of tag-question subjects with main sentence subjects.
Even if there is a complement clause with an available subject, ordinarily a tag-question subject will not agree with it. So, for example, compare
We regret that the king is dead, don't we?
*We regret that the king is dead, isn't he?
But after a sentence introduction used adverbially, it works the other way:
?*I believe the king is dead, don't I?
I believe the king is dead, isn't he?
By this test, however, your examples appear to have the same complement structure that we always thought they had.
