Первый лег уже начался.

No. Here лег is a noun, a borrowing from English leg, used in sports and games for a leg, round, or one part of a multi-part match.

It is not the verb лёг meaning lay down or went to bed.

This can be confusing because Russian often writes ё as е in ordinary text. So лёг may appear as лег in some contexts. But in this sentence, первый лег clearly works as a noun phrase: the first leg.

Because лег is a masculine singular noun, and первый has to agree with it.

So:

• первый = masculine singular
• первая = feminine singular
• первое = neuter singular

Since the noun is лег, the correct form is первый лег.

It is in the nominative case because it is the subject of the sentence — the thing that has started.

So the structure is basically:

• первый лег = the first leg
• уже начался = has already started / already started

Because the verb here is начаться, not начать.

• начать = to start something
• начаться = to begin / to start (when the event itself is starting)

In this sentence, the subject is the first leg. The leg is beginning, so Russian uses начаться:

• лег начался = the leg started / began

If you used начал, it would suggest that someone started something, and the sentence would need a direct object.

Here -ся is part of the verb начаться and helps make it intransitive: to begin, to start.

It does not literally mean itself in a natural English translation here. So it is best to learn:

• начать = to start something
• начаться = to start, to begin

Compare:

• Он начал матч. = He started the match.
• Матч начался. = The match started.

Your sentence follows the second pattern.

Because Russian does not have a separate tense that works exactly like the English present perfect.

So Russian often uses the past tense where English uses either:

• started
• has started

In this sentence, уже makes has already started a very natural English translation:

• Первый лег уже начался.
• The first leg has already started.

In the Russian past tense, the verb agrees with the subject in gender and number.

Since лег is masculine singular, the verb is also masculine singular:

• начался = masculine singular
• началась = feminine singular
• началось = neuter singular
• начались = plural

So начался matches лег.

Уже means already.

It shows that the action has happened earlier than expected, or that it is now true at this point.

So:

• лег начался = the leg started / has started
• лег уже начался = the leg has already started

No, Russian word order is fairly flexible. This version is natural and neutral.

Other orders are possible, but they change the emphasis:

• Первый лег уже начался. = neutral
• Уже начался первый лег. = emphasizes already started
• Первый лег начался уже. = sounds more marked, often contrastive

So the sentence as given is a normal way to say it.

Russian has no articles, so there is no separate word for the or a/an.

Whether something is understood as the first leg or a first leg depends on context. In a real situation, this sentence would normally be understood as the first leg.

So:

• Первый лег can mean the first leg
• Russian does not need an article to express that