После первой недели мы начали переговоры о контракте и о зарплате.

Because после requires the genitive case.

• первая неделя = the first week (nominative, dictionary form)
• после первой недели = after the first week (genitive)

So both words change:

• первая → первой
• неделя → недели

This is a very common pattern in Russian:

• после урока = after the lesson
• после работы = after work
• после первой недели = after the first week

Because it is the feminine singular genitive form of the adjective первый.

The noun неделя is feminine, so the adjective must agree with it in:

• gender
• number
• case

Here is the pattern:

• первая неделя — nominative
• первой недели — genitive

So первой is simply the correct feminine singular genitive form to match недели.

Yes — and that is a very common source of confusion.

недели can mean two different things depending on context:

1. genitive singular of неделя

   • после первой недели = after the first week

2. nominative/accusative plural of неделя

   • недели прошли быстро = the weeks passed quickly

In your sentence, it must be genitive singular because it comes after после.

Начали is the past tense plural form of начать = to begin / to start.

The subject is мы = we, so the verb must be plural:

• я начал / начала = I began
• ты начал / начала
• он начал
• она начала
• мы начали
• они начали

So мы начали means we began or we started.

Russian often expresses this idea as начать переговоры = to begin negotiations.

This highlights the start of the process, not the whole process itself.

Compare:

• мы начали переговоры = we began negotiations
• мы вели переговоры = we were conducting negotiations
• мы договорились = we reached an agreement

So начали переговоры is very natural if the sentence is about when the talks started.

In Russian, переговоры is normally used in the plural when it means negotiations / talks.

So:

• переговоры = negotiations
• вести переговоры = to conduct negotiations
• начать переговоры = to begin negotiations

Using a singular form here would not sound natural for this meaning. This is one of those nouns that Russian prefers in the plural.

Because the preposition о means about / concerning, and it requires the prepositional case.

So:

• контракт → о контракте
• зарплата → о зарплате

This is a standard pattern:

• говорить о работе = to talk about work
• думать о семье = to think about family
• переговоры о контракте = negotiations about the contract

Yes, о контракте и зарплате is also possible.

Russian often repeats the preposition for clarity or emphasis:

• о контракте и о зарплате
• о контракте и зарплате

Both are grammatical.

Repeating о can make the sentence sound a little clearer, especially when the speaker wants to separate the two topics distinctly:

• the contract
• the salary

So in this sentence, the repetition is natural and not strange.

Both can often be translated as contract, but there can be a difference in usage.

• договор is a very common general word for an agreement or contract
• контракт often sounds a bit more formal, businesslike, or specific, and is also common in employment/business contexts

In many contexts they overlap. Here контракт is perfectly natural if the topic is an employment or business contract.

Зарплата means pay, salary, or wages, depending on context.

It is a very common everyday word for what someone is paid for their work.

Some related words:

• зарплата = salary/pay
• оклад = fixed base salary
• заработная плата = the full formal term for salary/wages

So о зарплате means about salary/pay.

Yes, that word order is also possible:

• После первой недели мы начали переговоры...
• Мы начали переговоры после первой недели...

Both are correct.

Russian word order is more flexible than English, because case endings show the grammatical roles. The choice often depends on emphasis.

• После первой недели at the beginning emphasizes the time frame: After the first week, ...
• Мы начали переговоры first puts the focus more on the action itself

So the original sentence sounds very natural, especially if the speaker wants to set the time first.

Not quite.

• после первой недели = after the first week
• через неделю = in a week / one week later

These can be close in some situations, but they are not identical.

После первой недели refers to a point after that first week has passed.

Через неделю usually means one week later from some reference point.

So the original phrase is more specifically tied to the first week as a period.

Russian has no articles like a or the.

So контракт by itself does not automatically tell you whether it is:

• a contract
• the contract

You understand that from context.

In this sentence, о контракте could mean either about the contract or about a contract, depending on the situation already known to the speakers. Very often, the wider context makes it clear.