Командный зачёт важен для интеллектуальной игры.

It is a set phrase used in competitions, sports, quizzes, and similar events. Командный зачёт means the team standings, team score, or team classification as opposed to individual results.

So the idea is not just a team, but the way results are counted for the team as a whole.

Yes — зачёт has several meanings in Russian.

Common meanings include:

• a pass/credit in an educational setting
• credit/recognition
• in competitions, counting toward the standings or the standings/category itself

In this sentence, it has the competition-related meaning. So here зачёт is about how results are counted or the final team ranking/category, not an exam pass.

In Russian, the verb to be is usually omitted in the present tense.

So:

• Командный зачёт важен
  literally looks like Team standings important
• but it means The team standings are important

This is completely normal Russian grammar.

Because важен is the short form of the adjective важный.

In Russian:

• the full form is often used before a noun:
  важный вопрос = an important question
• the short form is often used in the predicate, meaning is important:
  вопрос важен = the question is important

So in this sentence:

• Командный зачёт важен = The team standings are important

Using важный here would be unnatural in standard Russian.

Because the subject is зачёт, and зачёт is masculine singular.

The short-form adjective has to agree with the noun:

• masculine singular: важен
• feminine singular: важна
• neuter singular: важно
• plural: важны

So:

• зачёт → важен

Because командный agrees with зачёт.

Since зачёт is:

• masculine
• singular
• nominative

the adjective must also be:

• masculine
• singular
• nominative

So we get:

• командный зачёт

If the noun changed, the adjective would change too.

Because the preposition для requires the genitive case.

So:

• dictionary form: интеллектуальная игра
• after для: интеллектуальной игры

Both words change because the adjective must agree with the noun:

• интеллектуальная → интеллектуальной
• игра → игры

So для интеллектуальной игры means for an intellectual game or for the intellectual game.

Literally it is an intellectual game, but in natural usage it often means something like:

• a quiz game
• a trivia game
• a knowledge-based competition
• a mind game or brain game in some contexts

In Russian, this phrase is commonly used for organized games where knowledge, logic, or quick thinking matter.

Yes. Russian word order is relatively flexible.

The neutral order here is:

• Командный зачёт важен для интеллектуальной игры.

But other orders are possible if you want to emphasize something different, for example:

• Для интеллектуальной игры важен командный зачёт.

That puts more focus on for an intellectual game.

The basic meaning stays similar, but the emphasis changes.

You could, but the meaning would change.

• важен для интеллектуальной игры = important for an intellectual game
• важен в интеллектуальной игре = important in an intellectual game

So:

• для shows relevance or importance for the game as a whole
• в places the importance inside the context of the game

The original sentence with для sounds more like team standings matter for this type of game. Using в would sound more like within the game, team standings are important.