Тренер дал команду, и мы начали бегать.

Дал is the perfective past of дать and means gave as a completed action. In this sentence, the coach gave one command, and that action is treated as a single finished event.

Давал would be imperfective and would suggest something like:

• he was giving a command,
• he used to give commands,
• or the action is viewed as ongoing/repeated rather than completed.

Here, the completed action gave the command fits best, because it is followed by the result: we started running.

Because дать normally takes a direct object in the accusative case, and команда is a feminine noun.

Its forms are:

• nominative: команда
• accusative: команду

So:

• Тренер дал команду = The coach gave a command

This is very similar to English give an order/command.

Because the subject is мы = we.

Russian past tense agrees with the subject in gender/number:

• начал = he started
• начала = she started
• начало = it started
• начали = they / we started

Since мы is plural, the verb must be начали.

After начать, Russian usually uses the imperfective infinitive, because the focus is on the beginning of an activity, not on its completion.

So:

• начали бегать = started running

Here, бегать is the imperfective verb meaning to run as an activity, often without emphasizing one specific destination or one single completed run.

Бежать often refers more to running in one direction / being in the process of running somewhere.

So:

• начали бегать = began the activity of running
• начали бежать would sound more like started to run off / started running in a specific direction, depending on context

In sports or training contexts, начали бегать is very natural.

This is a very common Russian verb pair.

• бежать = to be running, to run in one direction, a concrete motion
• бегать = to run habitually, repeatedly, or as a general activity

Examples:

• Я бегу домой. = I am running home.
• Я люблю бегать по утрам. = I like running in the mornings.

In your sentence, the meaning is about starting the activity, so бегать fits well.

Because this sentence contains two clauses with their own verbs:

1. Тренер дал команду
2. мы начали бегать

Russian normally uses a comma before и when it connects two independent clauses.

So the comma is there for the same basic reason as in English:

• The coach gave the command, and we started running.

Yes. That would also be correct, and it would make the recipient explicit:

• Тренер дал команду = The coach gave a command.
• Тренер дал нам команду = The coach gave us a command.

The original sentence does not explicitly say to us, but it is understood from the context, especially because the next clause says we started running.

Russian has no articles like a or the.

Whether команду means a command or the command depends on context.

So:

• Тренер дал команду can mean The coach gave a command
• or The coach gave the command

Usually the situation tells you which one is meant.

Yes. Russian word order is fairly flexible, though not completely free. The original order is very natural and neutral:

• Тренер дал команду, и мы начали бегать.

Other orders are possible for emphasis, for example:

• Команду дал тренер, и мы начали бегать.
  This emphasizes it was the coach who gave the command.
• И мы начали бегать, когда тренер дал команду.
  This restructures the sentence and changes the rhythm/focus.

For a learner, the original version is the best neutral model.

Тренер is in the nominative case, because it is the subject of the sentence.

It is the person performing the action:

• Тренер дал команду = The coach gave the command

The accusative object is команду, the thing being given.

It can be translated in several ways depending on context.

In this sentence, дал команду often means:

• gave the command
• gave the order
• gave the signal

In a sports or training context, English might naturally say:

• The coach gave the signal, and we started running or
• The coach gave the command, and we started running

So the Russian phrase is broad enough to cover those ideas.

Yes, in many contexts you could, and the meaning would be similar.

• начали бегать = started running
• стали бегать = began to run / started running

However, начать is often the more direct and neutral way to say to start doing something.

Стать + infinitive is also common, but sometimes it can sound slightly more like came to do / took up doing / started to depending on context.

In this sentence, начали бегать is the most straightforward choice.