How many syllables does the word there have?

Answer 1

There is only one syllable.
The word is comprised of: [th]-'A'-r => [th]-'A'-'E'-r, depending on speaker.
[The pronunciation of the letters is by their name-- "A" = ay, the letter A]

In some dialects it may be possible to pronounce this as two syllables, but I think it is unlikely:
"They"-"er"
If these two sounds blend together, though, it is just one long syllable.