What is the official 'name' for the 'there is' / 'there are' construction? Is it a verb phrase or a lexical verb?
I'd say possibly a verb but it must be the most difficult term to Google.
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It's the output of the English syntactic rule, transformation, or alternation There-Insertion.
There-insertion takes as input a sentence with one of a large set of verbs,
and generates as output a sentence meaning the same thing, but having a dummy there as subject. The original subject is moved to a position after the first auxiliary verb if there is one, or after one of many locative/existential predicates like stand or arise.
A commotion arose. ➞ There arose a commotion.
A giraffe is standing in the shower. ➞ There is a giraffe standing in the shower.
All the verbs that govern There-insertion are concerned with (coming into) existence and/or with location. The origin of there in this construction is the locative distal deictic there, and locatives are common in the construction, but dummy there is no longer the same word as demonstrative there:
As for what name there is/are has syntactically, that chunk really doesn't have one.
There is a dummy subject, and is/are an auxiliary verb -- though it's not the only verb that governs There-Insertion -- and they don't form a constituent; just a subject plus an auxiliary verb. That is the kind of structure that invites contraction, though, and that's why there's is so common, even with plural subjects -- once a structure has been contracted, it's frozen, and agreement won't work.
The construction is, as I said, the output of a transformation. Some people may call it an existential construction, but it's not the only one such in English, and that has to do with what it means, not with its grammar.
This is generally called an existential clause with there functioning as a kind of dummy subject.
For example, the version without the existential clause of 'There is someone who cares' is 'Someone who cares exists', and it is clear that semantically, 'someone who cares' is the subject.