Eg. There is no water in the Jug. Is "there" a demonstrative adjective in the example?

Answer 1

"There", as in "there is", is an adverb, which comes before certain verbs ("be", "seem", "appear", etc.).

"There", after the verb, is an adverb of place, as in "we didn´t want to go there" / "Who is there?".

"Demonstrative Pronouns" ("this", "these"; "that", "those") are used as subject of a verb, such as in the following sentence:
"This is exactly what I mean."
The same words ("this", "that", and their plurals") may function as Demonstrative Adjectives (= determiners) if they come before a noun:
"That book has just been returned."

But there is another word, "their" (pronounced like "there" in American English), which is a possessive adjective ("Their computer ...").

Answer 2

In your example "there" is a pronoun (it's the subject).

"There" can also be an adverb but not an adjective.