Нам пришлось дождаться автобуса.

Because this sentence uses an impersonal construction with прийтись.

With прийтись / приходиться meaning to have to, the person affected is normally put in the dative:

• мне пришлось = I had to
• тебе пришлось = you had to
• нам пришлось = we had to

So нам does not mean to us literally here in normal English style; it marks the people who were forced by circumstances to do something.

Пришлось is the past tense of прийтись in the meaning to have to / to be forced to / to end up having to.

So:

• Нам пришлось... = We had to...

It often suggests that the situation made it necessary, not that someone simply chose it.

That is an important nuance:

• Нам пришлось дождаться автобуса = circumstances made us wait until the bus came
• it can sound a bit like we had no choice

Because пришлось is being used impersonally.

There is no grammatical subject like мы here. Russian is not saying we had in the same structure English uses. Instead, it is more like:

• to us it happened that it was necessary to...

In impersonal past constructions, Russian often uses the neuter singular form:

• мне пришлось
• ему пришлось
• нам пришлось

So пришлось does not agree with нам.

Because дождаться means to wait until something/someone arrives or happens, with a clear result.

Compare:

• ждать = to wait
• дождаться = to wait long enough for something to finally come/happen

So:

• Нам пришлось ждать автобус = We had to wait for the bus
  
  • focus on the process of waiting
• Нам пришлось дождаться автобуса = We had to wait until the bus came
  
  • focus on reaching the result

In your sentence, the speaker presents the waiting as something that reached its endpoint: the bus finally came.

Because дождаться governs the genitive.

So:

• дождаться кого? чего?
• дождаться автобуса
• дождаться друга
• дождаться ответа

This is just something you need to learn with the verb.

A useful pattern:

• ждать автобус or ждать автобуса can both occur
• but дождаться normally takes genitive, so дождаться автобус would be wrong

The difference is mainly aspect and result.

• Нам пришлось ждать автобус
  = We had to wait for the bus
  This focuses on the activity/process.

• Нам пришлось дождаться автобуса
  = We had to wait until the bus came
  This focuses on the completed outcome.

So дождаться implies that the waiting was successful: the bus eventually arrived.

No. That is incorrect.

With пришлось in this meaning, Russian normally uses the dative, not a nominative subject:

• нам пришлось = we had to

So you should say:

• Нам пришлось дождаться автобуса

not:

• Мы пришлось...

If you want a structure with мы, you would need a different expression, for example:

• Мы были вынуждены дождаться автобуса = We were forced to wait for the bus
• Мы должны были дождаться автобуса = We were supposed to / had to wait for the bus

They are similar, but not identical.

• Нам пришлось дождаться автобуса
  = We had to wait for the bus / We ended up having to wait for the bus
  This stresses circumstances and often sounds more natural for an unwanted necessity.

• Мы должны были дождаться автобуса
  = We were supposed to wait for the bus / We were required to wait for the bus
  This can sound more like obligation, duty, or expectation.

So пришлось often means:

• we had no practical choice

while должны были often means:

• we were expected or required to do it

Yes. Russian word order is flexible.

The neutral order here is:

• Нам пришлось дождаться автобуса.

But other orders are possible for emphasis, for example:

• Автобуса нам пришлось дождаться.
• Нам автобуса пришлось дождаться.

These alternatives may emphasize автобуса or sound more marked in context.

For a learner, the given version is the most neutral and safest to use.

Yes, дождаться is perfective.

That makes sense because the sentence refers to a completed whole event: they waited, and the bus finally came.

After пришлось, either imperfective or perfective infinitives can appear, depending on meaning:

• пришлось ждать = had to wait
  
  • focus on process
• пришлось дождаться = had to wait until it arrived
  
  • focus on successful completion/result

So the perfective verb here is very natural because the action reached its endpoint.

They are:

• прийтись → past form here: пришлось
• дождаться → infinitive form appears directly in the sentence

Useful related forms:

For прийтись:

• приходится = have to, am forced to
• придётся = will have to
• пришлось = had to

For дождаться:

• дождусь, дождёшься... = I will wait until..., you will wait until...
• it is perfective, so it does not normally have a simple present meaning

The stress is:

• Нам пришлОсь дождАться автОбуса.

Approximate pronunciation:

• нам приш-ЛОС' даж-ДАЦ-ца аў-ТО-бу-са

A few notes:

• пришлось has stress on the last syllable: -лось
• дождаться has stress on -да-
• автобуса has stress on то

Also, in natural speech, дождаться is pronounced with -ться sounding roughly like ца.