En la plaza hay una fuente enorme rodeada de palmeras.

Use hay when you want to introduce or point out the existence of something: “there is/are.”
By contrast, estar (e.g. La fuente está en la plaza) locates or describes something already known.
So Hay una fuente en la plaza focuses on the fact that the fountain exists there, not just where it is.

In Spanish, descriptive adjectives typically follow the noun: fuente enorme is the neutral, most common order.
You can say una enorme fuente, but placing the adjective before the noun adds a touch of emphasis or stylistic flair, almost like “what a huge fountain!”

Yes. Rodeada de palmeras is a reduced relative clause (a past participle phrase functioning as an adjective).
It’s equivalent to saying que está rodeada de palmeras, but Spanish often drops que está for brevity:
“Hay una fuente enorme [que está] rodeada de palmeras.”

With rodear (to surround), Spanish uses rodear de + [things] to indicate what’s doing the surrounding.
Thus rodeada de palmeras = “surrounded by palm trees.”

Adjectives (including past participles used adjectivally) must agree in gender and number with the noun they modify.
Here, fuente is feminine singular, so rodeada takes the feminine-singular ending “-a.”

Absolutely. You could say:
“Hay una fuente enorme rodeada de palmeras en la plaza.”
Fronting En la plaza makes the location your speech’s starting point; moving it to the end shifts the focus onto the fountain first, then names its location.