Мы дождались друга.

Because дождаться requires the genitive case: дождаться кого? чего?
So:

• друг = nominative, dictionary form
• друга = genitive singular

This is one of the key things to memorize with this verb: дождаться кого-то / чего-то.

For an English speaker, this can feel strange because English says wait for a friend, while Russian changes the noun ending instead of using a preposition like for.

Дождались is the past tense of дождаться, which means something like:

• to wait until someone/something arrives
• to finally get to see/receive someone or something after waiting

It is perfective, so it emphasizes a completed result.

So:

• Мы ждали друга = We were waiting for a friend / our friend
• Мы дождались друга = We waited, and in the end the friend arrived / we finally got to him

In other words, дождались includes the idea of successful waiting or waiting to the point of completion.

The -ся here is part of the verb дождаться. It is not literally reflexive in the sense of we waited ourselves.

In Russian, -ся often creates a new verb with its own meaning. So it is best to learn дождаться as a whole vocabulary item, not as ждать + ся with a transparent word-for-word meaning.

So:

• ждать = to wait
• дождаться = to wait until, to finally get after waiting

This is a very common pattern in Russian: a prefix and/or -ся can produce a verb whose meaning must be learned as a unit.

Because the subject is мы, so the verb is in the past plural form.

Russian past tense agrees with the subject in number, and in the singular it also agrees in gender.

Here are the past forms of дождаться:

• я дождался / дождалась
• ты дождался / дождалась
• он дождался
• она дождалась
• мы дождались
• вы дождались
• они дождались

So -и shows that the subject is plural.

Yes, but it means something different.

• Мы ждали друга focuses on the process of waiting.
• Мы дождались друга focuses on the result: the waiting ended successfully.

Compare:

• Мы ждали друга два часа. = We waited for our friend for two hours.
  This does not by itself say whether the friend ever came.

• Мы дождались друга. = We waited for our friend, and he finally came / we finally got to see him.

This is an important aspect contrast in Russian.

Russian has no articles, so nouns do not automatically show a or the.

That means друга can be understood from context as:

• a friend
• the friend
• sometimes our friend, if the context makes that clear

If you want to say our friend explicitly, Russian would normally say:

• Мы дождались нашего друга.

So the sentence Мы дождались друга does not literally contain the word our. If an English translation uses our friend, that meaning is coming from context, not directly from the Russian words alone.

No. Russian word order is fairly flexible, and changing it usually changes emphasis, not the basic meaning.

• Мы дождались друга. = neutral
• Друга мы дождались. = emphasizes friend
• Мы друга дождались. = can sound more contrastive or conversational

The most neutral version for a learner is the original:

Мы дождались друга.

So yes, the order can change, but the original sentence is a good standard model to learn first.