Cuando empieza a llover, todos corremos adentro para no mojarnos.

Because this sentence describes a habitual or repeated situation: whenever it starts raining, we all run inside. In Spanish, that kind of general pattern is normally expressed with the present tense.

So:

• Cuando empieza a llover... = When / Whenever it starts to rain...

If you were talking about one specific past moment, you would change the tense:

• Cuando empezó a llover... = When it started to rain...

This is the very common structure empezar a + infinitive, which means to begin/start to do something.

So:

• empieza a llover = it starts to rain
• empieza a correr = he/she starts to run
• empezamos a comer = we start eating

The a is just part of the verb pattern after empezar in this meaning.

Because llover is usually an impersonal verb in Spanish. It works like to rain in English: we do not usually say who is doing the raining.

So Spanish normally says:

• Llueve = It’s raining
• Empieza a llover = It’s starting to rain

The it in English does not refer to a real thing, and Spanish simply leaves that kind of subject out.

Because corremos means we run, and in this sentence todos is understood as all of us.

So the idea is:

• todos corremos = we all run

If you said:

• todos corren

that would mean they all run or everyone runs, without including the speaker in the same way.

This is an important point: todos does not automatically force a third-person verb. The verb depends on who todos refers to.

Yes, but it is not necessary.

Spanish often leaves out subject pronouns because the verb ending already shows the subject:

• corremos already means we run

So:

• Todos corremos adentro...
• Nosotros corremos adentro...

Both are possible, but the version without nosotros sounds more natural unless you want extra emphasis or contrast.

Yes.

You could simply say:

• Cuando empieza a llover, corremos adentro para no mojarnos.

That still means When it starts to rain, we run inside so we don’t get wet.

Adding todos emphasizes that all of us do this, not just some of us.

Adentro means inside or indoors.

So:

• corremos adentro = we run inside

It is an adverb of place. In many contexts, Spanish speakers might also use dentro. In Spain, dentro is often very common, but adentro is perfectly understandable.

Para often means for, but here it introduces a purpose:

• para = in order to / so as to

When you add no, it becomes:

• para no = in order not to / so as not to

So:

• para no mojarnos = so as not to get wet

Because with an infinitive like mojarse, object and reflexive pronouns are usually attached to the end.

So:

• mojarnos = to get ourselves wet / to get wet

This comes from:

• mojarse = to get wet
• nos = ourselves / us

Together:

• mojar + nos → mojarnos

You do not say nos mojarnos.

Because mojar and mojarse are different.

• mojar = to wet something
• mojarse = to get wet

Examples:

• La lluvia moja la ropa. = The rain wets the clothes.
• Nos mojamos con la lluvia. = We get wet in the rain.

In your sentence, the meaning is so that we do not get wet, so the reflexive form mojarnos is the natural choice.

Because after para, Spanish normally uses the infinitive when the subject stays the same.

Here the subject of both actions is we:

• corremos = we run
• mojarnos = for us to get wet

So Spanish says:

• corremos adentro para no mojarnos
  we run inside so as not to get wet

If you used a fully conjugated verb like no nos mojamos, it would not fit this structure.