Mi atendos vin dum unu minuto.

Atendos means will wait.

It is built from:

• atend- = the root meaning wait
• -os = the future-tense verb ending

So:

• mi atendas = I wait / I am waiting
• mi atendis = I waited
• mi atendos = I will wait

Esperanto verbs always show tense with endings like this.

Because vin is the direct object form.

• vi = you as the subject
• vin = you as the object

In this sentence, mi is the one doing the waiting, and vin is the person being waited for.

The -n ending marks the direct object in Esperanto.

So:

• Vi atendos min. = You will wait for me.
• Mi atendos vin. = I will wait for you.

Because Esperanto atendi is a transitive verb. It directly takes an object.

English says:

• wait for someone

Esperanto says:

• atendi iun = literally wait someone, but naturally meaning wait for someone

So Mi atendos vin is the normal way to say I will wait for you.

You should not say atendi por vin here.

Dum means during or for the duration of.

In this sentence, dum unu minuto tells you how long the waiting will last: for one minute.

So:

• Mi atendos vin dum unu minuto. = I will wait for you for one minute.

This is a very common way to express duration in Esperanto.

Because dum expresses duration of time, while por usually expresses purpose, intended use, or benefit.

Compare:

• dum unu minuto = for one minute / for the duration of one minute
• por vi = for you
• por manĝi = to eat / for eating

So in this sentence, dum is correct because the idea is time length, not purpose.

Because minuto comes after the preposition dum, and nouns after a preposition normally do not take -n.

So:

• dum unu minuto = correct

A useful extra point: Esperanto can also express duration without a preposition by using the accusative:

• Mi atendos vin unu minuton.

That also means I will wait for you for one minute.

So both patterns exist:

• dum unu minuto
• unu minuton

But in your sentence, since dum is already there, minuto stays without -n.

No, Esperanto word order is fairly flexible, although Mi atendos vin dum unu minuto is the most neutral and natural order.

Because vin has the -n ending, you can still tell it is the object even if the order changes.

For example:

• Mi atendos vin dum unu minuto. = neutral
• Vin mi atendos dum unu minuto. = emphasizes you
• Dum unu minuto mi atendos vin. = emphasizes the time span

So the usual order is simple and clear, but other orders are possible for emphasis.

Normally, yes, you should include mi.

Esperanto verbs do not change according to person:

• mi atendos
• vi atendos
• li atendos

Since atendos by itself only tells you the tense, not the subject, the pronoun is usually needed.

So Mi atendos vin dum unu minuto is the normal full sentence.

Yes, dum minuto is possible, but dum unu minuto is more explicit: it clearly means for exactly one minute.

• dum unu minuto = for one minute
• dum minuto = for a minute, sometimes a little less explicit or more contextual

Using unu is very natural when you want to state a precise amount of time.

It can be used for both.

Examples:

• Mi atendas vin. = I am waiting for you.
• Mi atendas la buson. = I am waiting for the bus.
• Mi atendas respondon. = I am waiting for a reply.

So in this sentence, vin is just one possible direct object of atendi.