Learn Hardcore German: Das Spiel findet in einem kleinen Stadion am Rand der Stadt statt. - The game takes place in a small stadium at the edge of the city.

The basic (dictionary) form is stattfinden.
In a main clause, the conjugated part goes to position 2, and the prefix moves to the end of the clause.

So:

Infinitive:
- Das Spiel soll morgen stattfinden.
Present tense, main clause:
- Das Spiel findet morgen statt.

The prefix statt can never stand alone; it always belongs to finden here.
In the perfect tense the verb is no longer split:

Das Spiel hat gestern stattgefunden.

Could I just say Das Spiel ist in einem kleinen Stadion … instead of findet … statt?

You can say Das Spiel ist in einem kleinen Stadion, but there is a nuance:

findet statt = takes place / is held (focus on the event happening)
ist in = is in / is located in (more neutral, describes location)

For events (games, concerts, meetings) stattfinden is the most idiomatic verb:

Das Konzert findet in der Stadthalle statt.
Die Besprechung findet morgen um neun Uhr statt.

Das Spiel ist in einem kleinen Stadion is grammatically fine, but German speakers will naturally prefer Das Spiel findet in einem kleinen Stadion statt when talking about where the event is held.

Why is it in einem kleinen Stadion and not in ein kleines Stadion?

Because in here expresses location (where something is), not direction (where something is going to).

The preposition in can take either:

Dative for location (answering Wo? – where?)
Accusative for direction (answering Wohin? – where to?)

In your sentence:

Wo findet das Spiel statt? – In einem kleinen Stadion. → Dative

Compare:

Das Spiel findet in einem kleinen Stadion statt.
(It takes place in a small stadium – location → dative: *einem)*
Die Mannschaft geht in ein kleines Stadion.
(The team goes into a small stadium – direction → accusative: *ein)*

Why is the adjective kleinen and not kleines or kleiner?

Kleinen is the adjective in dative singular after an indefinite article (einem).

We have:

Stadion – neuter noun (das Stadion)
The phrase is in einem … Stadion → dative singular neuter
Dative singular (with any article) normally gives the adjective ending -en

So:

Nominative:
- ein kleines Stadion (subject)
Accusative:
- in ein kleines Stadion (direction)
Dative:
- in einem kleinen Stadion (location)

General rule: with an article, almost all dative singular adjectives end in -en:

mit einem guten Freund
bei einer netten Frau
aus einem alten Haus

Why einem and not just ein before Stadion?

Again, this is because of the dative case.

The indefinite article ein changes with case and gender:

Neuter (Stadion)
- Nominative: ein Stadion
- Accusative: ein Stadion
- Dative: einem Stadion

Your preposition in (with location) requires the dative, so you must say:

in einem kleinen Stadion (not in ein kleines Stadion)

Similarly:

mit einem Freund (not mit ein Freund)
aus einem Land (not aus ein Land)

What exactly does am Rand mean, and why not an dem Rand?

Am is just the contraction of an dem:

an dem Rand → am Rand

So am Rand literally means “at/on the edge/border”.

In everyday German you almost always use the contracted form:

am Rand der Straße (at the edge of the street)
am Tisch (at the table)
am Fenster (at the window)

An dem Rand der Stadt is grammatically correct but sounds overly formal/unnatural in this context. Am Rand der Stadt is the normal, idiomatic phrase.

What case is der Stadt in am Rand der Stadt, and why?

Der Stadt here is genitive singular, not dative.

Basic noun: die Stadt (feminine)
Genitive singular: der Stadt

The structure is:

am Rand der Stadt
= an dem Rand der Stadt
→ Rand is a noun that takes a genitive complement: the edge of the city

So it literally means: “at the edge of the city.”

More examples with genitive after a noun:

das Ende des Films (the end of the film)
die Mitte der Woche (the middle of the week)
die Farbe des Autos (the color of the car)

Could I also say am Rand von der Stadt instead of am Rand der Stadt?

You can hear am Rand von der Stadt in spoken, colloquial German, but in standard written German the genitive am Rand der Stadt is preferred and sounds more natural.

Options:

Standard / neutral:
- am Rand der Stadt
Colloquial / more spoken:
- am Rand von der Stadt

What you shouldn’t say is am Rand von Stadt (you still need the article in German).

Why are Spiel, Stadion, Rand, and Stadt all capitalized?

In German, all nouns are capitalized, regardless of their position in the sentence.

So:

das Spiel – the game
das Stadion – the stadium
der Rand – the edge
die Stadt – the city

This is a core spelling rule in German and applies even when the noun is not at the beginning of the sentence:

Ich habe ein neues Auto.
Dieses Auto ist sehr schnell.

Why is it das Spiel – how do I know Spiel is neuter?

Spiel is simply listed in the dictionary as das Spiel (neuter).
For many German nouns, the gender must be memorized; there is no reliable rule that always works.

Some endings give clues (e.g. -ung → usually feminine, -chen → usually neuter), but Spiel doesn’t have a strong pattern; you just learn it as:

das Spiel (singular)
die Spiele (plural)

You get used to gender by seeing and using nouns with their articles:

das Spiel
das Stadion
die Stadt
der Rand

Why is the verb findet in second position, and why does statt go all the way to the end?

In a normal German main clause:

Something occupies the first position (often the subject).
The conjugated verb is in second position.
Other elements (objects, adverbials, etc.) come after that.
For separable verbs, the prefix goes to the very end of the clause.

In your sentence:

Das Spiel – position 1
findet – finite verb in position 2
in einem kleinen Stadion am Rand der Stadt – middle part
statt – separable prefix at the end

You can move other parts to the front for emphasis, but the conjugated verb still stays in second position and statt still goes to the end:

In einem kleinen Stadion findet das Spiel am Rand der Stadt statt.
Am Rand der Stadt findet das Spiel in einem kleinen Stadion statt.

How would I negate this sentence? Where does nicht go?

It depends on what you want to negate.

General negation of the whole statement (the game doesn’t take place there):
- Das Spiel findet nicht in einem kleinen Stadion am Rand der Stadt statt.
Here nicht stands before the whole prepositional phrase.
Negating specifically “in a small stadium” (maybe it’s in a big stadium there):
- Das Spiel findet nicht in einem kleinen Stadion, sondern in einem großen Stadion am Rand der Stadt statt.
Negating the stadium itself (using kein):
- Das Spiel findet in keinem kleinen Stadion am Rand der Stadt statt.

For a simple “no, this is not the place,” the most natural is:

Das Spiel findet nicht in einem kleinen Stadion am Rand der Stadt statt.

How do you pronounce Stadion and why is the st like “sht”?

Stadion is pronounced approximately:

[ˈʃtaːdi̯ɔn] → like “SHTAH-dee-on”

In standard German, st at the beginning of a root syllable (especially at the beginning of a word) is usually pronounced [ʃt] (“sht”):

stehen – [ˈʃteːən]
Straße – [ˈʃtraːsə]
Stadt – [ʃtat]

But when st comes inside a word after another consonant, it is usually [st]:

Abstand – [ˈapʃtant] (first st as [ʃt])
Lastwagen – [ˈlastvaːgən] (st here as [st])

So Stadion starts with the typical German “sht” sound.

What is the plural of Stadion, and does anything in the sentence change?

The usual plural of Stadion is Stadien:

das Stadion → die Stadien

Your sentence in the plural (talking about several games) could be:

Die Spiele finden in kleinen Stadien am Rand der Stadt statt.

Changes:

Das Spiel → Die Spiele
in einem kleinen Stadion → in kleinen Stadien
(plural, so no article needed; adjective in plural dative with no article also takes -en)

Everything else stays the same.