Nous n’avons personne à appeler maintenant, alors nous pouvons nous asseoir et boire un café.

Because personne is already part of the negative structure.

In French, ne ... personne means no one / nobody:

• Nous n’avons personne = We have nobody
• Je ne vois personne = I see nobody

So you normally do not add pas as well.
Using ne ... pas personne would be nonstandard in normal French.

Here, personne means nobody / no one.

That can be confusing, because personne can also mean person in other contexts.

Compare:

• Il y a une personne dehors. = There is a person outside.
• Il n’y a personne dehors. = There is nobody outside.

So in your sentence, because it appears with ne, it has the negative meaning: nobody.

French shortens ne to n’ before a vowel sound.

Since avons begins with a, you get:

• ne avons → n’avons

This is called elision and happens very often:

• je aime → j’aime
• ne est → n’est
• me appelle → m’appelle

In this structure, à + infinitive describes what there is or is not to do with a person or thing.

So:

• personne à appeler = nobody to call
• rien à faire = nothing to do
• quelqu’un à voir = someone to see
• un livre à lire = a book to read

It is a very common French pattern.
English often uses just to + verb, while French often uses à + infinitive after a noun or pronoun like this.

Because it depends on personne à ... and describes an action in a general, unfinished way.

In personne à appeler, the infinitive works like to call in English:

• nobody to call
• nothing to eat
• someone to ask

So appeler is not being conjugated for a subject here. It is just naming the action.

The first nous is the subject pronoun: we.

The second nous is the reflexive pronoun, because s’asseoir is a reflexive verb in this meaning:

• nous pouvons = we can
• nous asseoir = sit down / seat ourselves

So the whole thing literally works like:

• We can seat ourselves

French often uses reflexive verbs where English does not:

• se lever = to get up
• se reposer = to rest
• s’asseoir = to sit down

Both are related, but they appear in different contexts.

The dictionary form is s’asseoir.
When you use it after a conjugated verb, the reflexive pronoun changes to match the subject:

• je vais m’asseoir
• tu vas t’asseoir
• nous pouvons nous asseoir
• ils veulent s’asseoir

So in your sentence, the subject is nous, so the reflexive pronoun must also be nous.

Because after a conjugated modal verb like pouvoir, the next verb normally stays in the infinitive.

So:

• nous pouvons nous asseoir = we can sit down
• nous pouvons boire = we can drink

This is the same idea as in English:

• can sit
• can drink

Only pouvons is conjugated here. The other verbs stay in their base form.

Here, alors means so or therefore.

It links the two ideas:

• We have nobody to call now, so we can sit down and have a coffee.

Depending on context, alors can also mean then, but in this sentence it clearly works as a logical connector: so.

Both are possible in French, but they mean slightly different things.

• boire un café = to have a coffee / drink a cup of coffee
• boire du café = to drink coffee in a more general sense

In your sentence, un café sounds natural because it suggests one coffee, like sitting down for a coffee break.

Maintenant means now, and French adverbs can sometimes move around more flexibly than in English.

Here, personne à appeler maintenant means nobody to call now / right now.

This placement sounds natural and attaches maintenant to the idea of calling.
Other placements may also be possible depending on emphasis, for example:

• Maintenant, nous n’avons personne à appeler...
• Nous n’avons maintenant personne à appeler...

But the original sentence is smooth and idiomatic.

Yes. In everyday spoken French, many speakers would naturally say on instead of nous:

• On n’a personne à appeler maintenant, alors on peut s’asseoir et boire un café.

This is very common in conversation.
Your original sentence with nous is completely correct, but it sounds a bit more careful or formal than casual spoken French.

S’asseoir is the normal verb for to sit down.

French can also use se poser or other expressions in very informal speech, but s’asseoir is the standard and correct choice here.

A small nuance:

• s’asseoir = to sit down
• être assis = to be seated / to be sitting

So your sentence focuses on the action of sitting down, not just the state of already being seated.