Der Spielzug ist kurz, doch die Spielregel ist kompliziert.

Because German nouns have grammatical gender, and each noun has its own article:

• der Spielzug = masculine
• die Spielregel = feminine

There is not always a logical reason you can see from the meaning alone, so you usually need to learn the noun together with its article.

A helpful thing here is that Regel is a feminine noun, so compounds ending in -regel are also feminine. In German compounds, the last part usually decides the gender:

• die Regel → die Spielregel
• der Zug → der Spielzug

In German, all nouns are capitalized, not just proper names.

So in this sentence:

• Spielzug is a noun
• Spielregel is a noun

That is why both begin with a capital letter.

German very often builds new nouns by combining smaller nouns.

Here:

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

So:

• Spielzug = a game move / move in a game
• Spielregel = game rule / rule of the game

A useful rule: in German compounds, the last word is the main word, and the earlier part just describes it.

So:

• Spielzug is a kind of Zug
• Spielregel is a kind of Regel

This is a normal German main clause. The basic pattern is:

subject + verb + rest

So:

• Der Spielzug = subject
• ist = verb
• kurz = complement

German main clauses usually place the finite verb in second position. The same happens in the second clause:

• die Spielregel = subject
• ist = verb
• kompliziert = complement

That is why both parts follow the same pattern.

Because doch here connects two main clauses:

• Der Spielzug ist kurz
• doch die Spielregel ist kompliziert

In German, when two full main clauses are joined by a coordinating conjunction like doch, a comma is normally used.

So the comma helps separate the two complete thoughts.

Here, doch means something like but, however, or yet.

So the sentence contrasts two ideas:

• the move is short
• the rule is complicated

In many cases, doch and aber are both possible. Here, doch sounds a little more formal or literary than aber.

So:

• Der Spielzug ist kurz, aber die Spielregel ist kompliziert.
• Der Spielzug ist kurz, doch die Spielregel ist kompliziert.

Both are grammatical. The version with doch simply has a slightly different tone.

Because here they are predicate adjectives, not adjectives placed directly before a noun.

Compare:

• Der Spielzug ist kurz.
• die komplizierte Spielregel

In Der Spielzug ist kurz, the adjective comes after ist and describes the subject. In that position, German adjectives usually do not take endings.

So:

• ist kurz
• ist kompliziert

But before a noun, they do take endings:

• der kurze Spielzug
• die komplizierte Spielregel

Both are in the nominative case, because both are the subjects of their clauses.

• Der Spielzug ist kurz.
  • Der Spielzug is the subject
• Die Spielregel ist kompliziert.
  • Die Spielregel is the subject

After sein (ist), German uses the nominative for the subject, and the adjective after it stays unchanged.

Yes, that is grammatically possible, but it changes the meaning slightly.

• Der Spielzug ... / die Spielregel ... sounds like you are talking about a specific move and a specific rule, or about them as known items in context.
• Ein Spielzug ... / eine Spielregel ... sounds more general, like a move and a rule in a broader sense.

So the original sentence with der and die feels more definite and context-based.

A few useful pronunciation points:

• Sp at the beginning of a word is usually pronounced shp
  • Spiel sounds roughly like shpeel
• ie is usually a long ee sound
  • Spiel sounds like speel, but with sh at the start
• z in German is pronounced ts
  • Zug sounds like tsoog
• regel has a long e at the start: roughly RAY-gel

Very roughly:

• Spielzug ≈ SHPeel-tsoog
• Spielregel ≈ SHPeel-ray-gel

These are only approximate English-style guides, but they can help at first.

Yes, kompliziert is a very common and natural word in German.

You will often see it in sentences like:

• Die Aufgabe ist kompliziert.
• Die Erklärung ist kompliziert.

It works very much like English complicated. In everyday speech, people may also use schwierig in some contexts, but the meaning is not always exactly the same:

• kompliziert = complicated, complex
• schwierig = difficult

So die Spielregel ist kompliziert specifically suggests that the rule itself is complex.