Vamos al estadio los domingos para ver jugar a las niñas y a los niños.

In Spanish, the preposition a + the masculine singular article el always contracts to al.
So:

• a + el estadio → al estadio

You cannot say a el estadio; it sounds incorrect. This contraction happens only with a + el and de + el (del), not with la, los, las (e.g. a la casa, a los parques are not contracted).

Spanish uses the definite article with days of the week to talk about habitual actions:

• el domingo = on Sunday (one specific Sunday)
• los domingos = on Sundays (every Sunday / on Sundays in general)

So Vamos al estadio los domingos means “We go to the stadium on Sundays (regularly / habitually).”

The simple present in Spanish is commonly used for:

• habits/routines: Vamos al estadio los domingos = We go to the stadium on Sundays (as a routine).

Iremos (future) would sound like you’re talking about a future plan, not a regular habit.
Vamos a ir also points to a specific future occasion. For habits, the plain present vamos is the natural choice.

The phrase para ver jugar literally means “in order to watch (them) play.”

• ver = to see / to watch
• jugar = to play

Spanish often uses ver + infinitive to say “watch someone do something”:

• ver bailar a alguien = to watch someone dance
• ver jugar a alguien = to watch someone play

Just para ver would be “in order to watch (something)” without saying what.
Just para jugar would mean “in order to play (ourselves),” which changes the meaning.

In Spanish, when a direct object is a person (or beloved animal), you normally use the personal a:

• Veo a Marta. – I see Marta.
• Vemos jugar a las niñas. – We watch the girls play.

So a las niñas and a los niños are direct objects that refer to people. That’s why the a is required.
You can either repeat the a or omit one:

• …ver jugar a las niñas y a los niños (very clear)
• …ver jugar a las niñas y los niños (also heard, but repeating is safer/clearer in careful speech).

Traditionally, los niños (masculine plural) can refer to a mixed group of boys and girls.
However, many speakers today prefer more explicitly inclusive language:

• las niñas y los niños = girls and boys (mentions both genders separately)

This avoids ambiguity and is seen as more inclusive or sensitive in many contexts, especially in education, public speaking, and formal writing.

You could say ver jugar a niñas y niños, and it would be understood.
However, there is a nuance:

• a las niñas y a los niños: sounds more specific, like a known/identifiable group (for example, the kids from a particular team or school).
• a niñas y niños: sounds a bit more general or indefinite (some girls and boys, in general).

In everyday speech, including las/los is more common and feels more natural here.

Both orders are possible:

• ver jugar a las niñas y a los niños
• ver a las niñas y a los niños jugar

The first (ver jugar a…) is more common and flows more naturally in Spanish when using ver + infinitive structures.
The second is grammatically correct but sounds a bit heavier and is less commonly used in everyday speech.

Yes, that’s possible:

• para ver jugar a las niñas y a los niños = to watch the girls and boys play
  (focus on the activity itself, a compact structure)
• para ver cómo juegan las niñas y los niños = to see how the girls and boys play
  (slightly more focus on the way / manner they play)

Meaning is very close, but cómo juegan adds a tiny nuance of “seeing in what way they play.”

After verbs of perception like ver, oír, sentir, Spanish often uses another bare infinitive directly:

• ver jugar, ver correr, oír cantar, oír llorar

You do not add a before that second verb.
The a goes before the person you’re seeing/hearing, not before the infinitive:

• Correct: ver jugar a las niñas
• Incorrect: ver a jugar las niñas