Consider the following sentences:

Over the mountains live the trolls.
Past the river lies the village.
Under the water swim the fish.

In all of these situations we have what I think is a sentence
beginning with a prepositional phrase, and then the subject and verb
are inverted.

The question is basically why.  Why are the subject and verb inverted,
even though it's not a question?

One cannot claim that it's merely because you started the sentence
with a prepositional phrase.  That doesn't necessarily make any
difference...

After going to the store, I returned home.
To win her love, he bought her roses.

It's weird.  The best way I can think of describing what's happening
is:  when you start a sentence with a prepositional phrase THAT HAS TO
DO WITH GEOGRAPHY, the subject and the verb are inverted.  Because if
you try to concoct sentences in which this happens, you'll notice that
the prepositional phrase you begin with has to have something to do
with land, distance, or terrain.

But WHY?!?!  Did English teachers lose a battle to geography teachers
hundreds of years ago, and we still must do this out of a mark of
respect or shame?

I'm totally baffled.

Here's what Webster's New World English Grammar Handbook says about
the situation (p. 194):

Occasionally, prepositional phrases mayb e used in somewhat unusual
ways, for example as subjects of sentences.  These usages are somewhat
rare, but the construction of the prepositional phrase remains the
same:

1.  Over the fence is a home run.
2.  In shallow water is the place to find minnows.

However, I suspect that Webster's, despite its vaunted name, got the
explanation wrong, at least in their second example:  I submit the
subject of the second sentence is "place," which becomes clear when
you rearrange the elements of the sentence:

The PLACE to find minnows IS in shallow water.

My second piece of evidence is that, in my first example above,
"lives" would sound all wrong, because "trolls" is the true subject --
whereas Webster's seems to be asserting that if a prepositional phrase
is your subject, the verb must be singular.

No doubt about that.  E.g.:

To live well is the goal of life.

Anyhow.  So my question . . .  Two parts, really:

What is the rule for this phenomenon, and why the rule?

If you can't answer the rule, can somebody at least tell me what this
rule or phenomenon is called, so that I can look it up in grammar
manuals on my own?  But I don't even know what this situation is
called!  How is this conundrum termed, etc.

This has puzzled me for years, so thanks for any help.

The inversion of subject and verb is an option that may suit the tone
of a piece of writing, but I don't think there is a "rule." Consider
your sample sentences:

Over the mountains live the trolls.
Past the river lies the village.
Under the water swim the fish.

These can easily be rewritten:

Over the mountains the trolls live.
Past the river the village lies.
Under the water the fish swim.

Or, to seem less stilted:

The trolls live over the mountains.
The village lies past the river.
The fish swim under the water.

Time magazine used to be known for its frequent use of unconventional
word order. The humorist Wolcott Gibbs once spoofed this in The New
Yorker, culminating in this glorious summation: "Backward ran the
sentences until reeled the mind." The pattern of inverting subjects
and verbs is sometimes called "Timespeak," in honor of Time magazine.

More recently, a similar phenomenon has been given a new name:
Yodaspeak, after the wise, gnomelike Jedi master from "Star Wars" of
whom it might be said "Construct a straight sentence can he not."

The word order in those examples is consistent with a more literary
than conversational style and is also more characteristic of older
writing than contemporary.  Those three sentences would be right at
home in a fairy tale, and you might use them today in, say, telling a
story to your grandchildren--"Right up to our doorway came the flood
waters" or "Into the steaming pot fell the wolf"--but you'd never see
a construction like that in, say, a technical manual or your average
general-audience magazine article.

The sentences also follow German word order, which requires inverting
the subject and verb if anything other than the subject comes first. 
Some of our grammatical rules as well as our lexicon reflect a kinship
between English and the Germanic languages.  You are onto something
when you mention that the initial phrases in your examples refer to
place (and I do agree with you on the second example in Webster's),
but you must also note that they are adverbial phrases modifying the
verb and that they are placed in close proximity to the verb.  The
same might be true of, say, adverbial phrases of time, as in the
following phrase from "Hark, the Herald Angels Sing":  "Late in time
behold him come."  Other examples:  "At dawn came a message."  "All
night howled the storm."  "In November began the rainy season." 
Again, it could occur with adverbial phrases of means--all still
prepositional phrases:  "By foulest deceit won he the crown." 
"Through great endurance and persistence came enlightenment."  "By
experience comes wisdom."  It is no longer a common construction, but
we can still parse it and understand it readily, especially when
written and not spoken.

Archae0pteryx

The the term "invert" used in various forms in both the question and
the comments is the correct one.  Just to make it explicit, this is
called "subject-verb inversion."  It occurs in many other situations
than the one you have noticed, and as the other commenters noted,
there aren't rules for right and wrong, it just may sound more or less
colloquial or natural or stilted or old-fashioned depending on the
construction of the rest of the sentence.  For lists of situations
where subject-verb inversion occurs, see:

Capital Community College Foundation Guide to Grammar and Writing
http://webster.commnet.edu/grammar/subjects.htm

Grammar Station Grammar Guide (seven pages indexed at the bottom of this page)
http://www.grammarstation.com/grammarguide/UIntermediate.html

TestMagic
http://www.testmagic.com/grammar/explanations/inversion.htm

I have noticed that play-by-play sports broadcasters often use a
different sort of inverted sentence form these days.  For example:

"Took a mighty swing, did Casey."

While reading the question, I was thinking: yeah, "inversion",
German,..., maybe commas, if no inversion ...
but you all got their first with the mostest  -  :)

Really interesting has proved to be this thread ;-)

While I've learned from all of the comments, IMHO efn-ga's seems to
answer the questions as it's ultimately been formulated: "what this
rule or phenomenon is called": subject-verb inversion.

More common in my mother language -Spanish- than in English, this
grammatical phenomenon is used for a poetic effect, or else to
emphasize a certain part of the discourse, as in my initial sentence,
remarking the idea of "interest" as more important than "thread".
Maybe not a very good example, I hope it was illustrative enough.