Since "there exists a dog with black fur" is okay, is it generally true that "there + 'intransitive verb' ..." also okay?
For example, can I say "There thrive trees in the park" to mean "Trees in the park thrive"?
I doubt this has much to do with intransitive or transitive verbs.
There is a tree in the park is a fine sentence. If you look at the ODO entry for there, you see it gets combined with be in virtually all cases.
In sentences with verbs that do not denote "existence", there seems unnatural. Your first example, there exists, is virtually synonymous with there is.
This even goes for a sentence like:
There are trees thriving in the park.
Thriving is attributive to trees, and the main idea of the sentence is still there exist trees (and they are in the park, and they are thriving).