Al mediodía cayó granizo durante diez minutos, pero luego salió un arcoíris precioso sobre el barrio.

Because al is a + el. The noun is el mediodía, and in time expressions Spanish often uses a:

• a las tres
• a medianoche
• al mediodía

So al mediodía is the natural way to say at noon / around midday.

Yes, granizó is also possible. Both are grammatical.

• cayó granizo = hail fell
• granizó = it hailed

The version with caer + weather noun is very common and sounds very natural in narration. Spanish also does this with other things:

• cayó nieve
• cayeron cenizas

So cayó granizo is a normal, idiomatic choice.

Because granizo is being used as a general weather substance, not as a specific, already identified thing. Spanish often leaves out the article in this kind of context:

• cayó nieve
• cae lluvia ácida
• cayó granizo

If you were talking about specific hail already mentioned, an article could appear in a different sentence, but here the bare noun is the normal choice.

Because the sentence presents completed events in sequence.

• cayó granizo durante diez minutos = a finished event with a clear duration
• luego salió un arcoíris = the next event in the story

The preterite is the normal tense for this kind of narration. If you used the imperfect, it would sound more like background description or an ongoing situation:

• caía granizo... = hail was falling...
• salía un arcoíris... = a rainbow was coming out / appearing...

That is not the main idea here. Here, the speaker is telling what happened, step by step.

Durante diez minutos gives the duration of the hailstorm.

In Spain, durante is the most neutral and standard choice for this meaning. Por diez minutos is much less natural in Spain and may sound influenced by English or more typical of some Latin American usage.

So for Spain Spanish, durante diez minutos is the best option here.

Because pero is the normal word for but when you are simply contrasting two statements.

• Cayó granizo..., pero luego salió un arcoíris...

Sino is usually used after a negative statement to correct or replace something:

• No cayó nieve, sino granizo.

So in your sentence, pero is correct because there is no previous negation to correct.

Yes. Luego and después can both mean then / afterwards here.

• pero luego salió un arcoíris...
• pero después salió un arcoíris...

Both are natural. Luego is very common in Spain and works especially well in storytelling.

Because salir also means to come out or to appear, especially with things seen in the sky or in nature.

Very common examples:

• salió el sol
• salieron las estrellas
• salió un arcoíris

So here salió un arcoíris is a very natural way to say that a rainbow appeared.

Because in Spanish, descriptive adjectives usually go after the noun.

• un arcoíris precioso
• una casa grande
• un día horrible

Putting precioso after arcoíris sounds neutral and natural. You could say un precioso arcoíris, but that sounds more literary or more emotionally expressive. The version in the sentence is the everyday default.

Yes. Arcoíris is normally masculine, so you say:

• un arcoíris
• el arcoíris

So the article un is exactly what you would expect here.

Sobre el barrio means over the neighborhood or above the neighborhood.

Sobre is natural when something is physically positioned above a place, especially in the sky. A rainbow is not in the neighborhood in a literal sense, so en el barrio would not fit as well here.

So:

• sobre el barrio = above/over the neighborhood

is the most natural choice.

The accent marks show the correct stress and, in some cases, the syllable separation.

• cayó and salió have an accent because the stress falls on the last syllable.
• mediodía and arcoíris have an accent to show that the í is stressed and forms a separate syllable.

Those accents are important because they tell you how to pronounce the words correctly.