Die Spielregel bestimmt den Spielzug.

Because they show different grammatical cases.

• die Spielregel is the subject, so it is in the nominative case.
• den Spielzug is the direct object, so it is in the accusative case.

In this sentence, the articles help you see who is doing the action and what is being affected by it.

A quick breakdown:

• die Spielregel = the rule
• bestimmt = determines
• den Spielzug = the move

So the structure is:

• The rule determines the move.

The articles tell you.

• die can be nominative feminine singular
• den is accusative masculine singular

That means:

• Spielregel is the thing performing the action
• Spielzug is the thing receiving the action

This is very useful in German, because case marking often makes the roles clearer than in English.

Because the subject is singular: die Spielregel.

The verb bestimmen means to determine. In the present tense, it changes depending on the subject:

• ich bestimme
• du bestimmst
• er/sie/es bestimmt
• wir bestimmen
• ihr bestimmt
• sie/Sie bestimmen

Since die Spielregel is grammatically she (sie) and singular, the correct form is bestimmt.

Because German very often forms compound nouns by joining words together.

Here:

• Spiel = game
• Regel = rule

So Spielregel literally means game rule, or more naturally rule of the game / game rule.

The same happens with:

• Spielzug = game move / move in a game
• Hausaufgabe = house + task = homework
• Fußballspieler = football + player = soccer player

In German, these are usually written as one word.

You usually have to learn noun gender together with the noun:

• die Spielregel = feminine
• der Spielzug = masculine

A useful tip: the last part of a compound noun usually decides the gender.

• die Regel is feminine, so die Spielregel is feminine
• der Zug is masculine, so der Spielzug is masculine

This rule works for many German compounds.

Because German main clauses usually follow the verb-second rule.

In a normal statement, the finite verb comes in the second position:

• Die Spielregel | bestimmt | den Spielzug

That is very standard German word order: subject + verb + object

You could move another element to the front, but the verb would still stay second:

• Den Spielzug bestimmt die Spielregel.

That version is less neutral, but still grammatical.

Yes, in this sentence bestimmt is a verb from bestimmen.

But learners often notice that bestimmt can also mean certainly or definitely in other contexts. For example:

• Das ist bestimmt richtig. = That is certainly right.

So the word bestimmt can be:

• a verb form: Die Regel bestimmt den Zug.
• an adverb: Er kommt bestimmt. = He is definitely coming.

Here it is clearly the verb, because it is the action of the sentence.

Yes. German often uses the singular to express a general idea.

So Die Spielregel bestimmt den Spielzug can be understood as a general statement like:

• The rule determines the move
• A game rule determines the move

This does not have to mean only one specific rule and one specific move in a real situation. It can also be a more abstract or generic statement.

Depending on context, German could also use the plural:

• Die Spielregeln bestimmen die Spielzüge.

But the singular is perfectly normal for a general principle.

Usually, yes. Spielzug means a move or play in a game.

It is made from:

• Spiel = game
• Zug = move, turn, pull, train, draft, etc. depending on context

In game contexts, Zug often means move or turn.
So Spielzug is a move made in the game.

Depending on the context, German might also simply use Zug by itself, especially in games like chess:

• ein Zug = a move

Not exactly, because the German sentence is singular:

• Die Spielregel = the rule
• not die Spielregeln = the rules

If the sentence meant The rules determine the move, it would be:

• Die Spielregeln bestimmen den Spielzug.

So the singular/plural matters here.

A few small things may help:

• ie in Die and Spiel- is a long ee sound.
• sp at the start of Spielregel and Spielzug is usually pronounced like shp in standard German.
• z in Spielzug is pronounced ts.

So Spielzug sounds roughly like shpeel-tsoog.

That said, the grammar of the sentence is more important than perfect pronunciation at first.