Это семейный бизнес.

In Russian, the present-tense verb to be is usually omitted.

So:

• Это семейный бизнес. = It/This is a family business.

There is no separate word for is here. This is completely normal Russian.

You would only see forms of быть (to be) in other tenses, for example:

• Это был семейный бизнес. = It was a family business.
• Это будет семейный бизнес. = It will be a family business.

Here это means something like this or it in the pattern This/It is ...

It is very commonly used to identify or classify something:

• Это школа. = This is a school.
• Это мой брат. = This is my brother.
• Это семейный бизнес. = This/It is a family business.

So in this sentence, это is not directly attached to бизнес as an adjective. It is standing on its own.

Because это and этот are doing different jobs here.

• это = a standalone word meaning this / it is
• этот = an adjective-like word meaning this, used directly before a masculine noun

Compare:

• Это семейный бизнес. = This/It is a family business.
• Этот бизнес семейный. = This business is family-run / this business is a family business.

In your sentence, это is not modifying бизнес. It is introducing the whole statement.

Because семейный is an adjective, and it must agree with бизнес.

бизнес is:

• masculine
• singular
• nominative

So the adjective must also be:

• masculine
• singular
• nominative

That gives семейный.

Compare with other nouns:

• семейная компания = a family company (feminine)
• семейное дело = a family business / family matter (neuter)
• семейные традиции = family traditions (plural)

Yes, бизнес is a masculine noun.

Even though it is a loanword, it behaves like a regular masculine noun ending in a consonant, so it declines:

• nominative: бизнес
• genitive: бизнеса
• dative: бизнесу
• accusative: бизнес
• instrumental: бизнесом
• prepositional: о бизнесе

So it is not indeclinable.

Because this is a basic present-tense identification sentence.

In Russian, when you say that something is something, the predicate noun phrase is often in the nominative in simple present-tense statements, especially with это:

• Это проблема. = This is a problem.
• Это хороший знак. = This is a good sign.
• Это семейный бизнес. = This is a family business.

So both the adjective and the noun stay in the nominative here.

Usually yes: it means a business owned or run by members of a family.

But there is one important nuance.

In English, family business can also mean a private matter belonging to the family, as in That’s our family business.

In Russian, семейный бизнес normally means the commercial sense: a family-run business.

For the private matter sense, Russian would usually use something else, for example:

• Это наше семейное дело. = That is our family business in the sense of our private matter

So be careful: the English expression has two meanings, but семейный бизнес usually has only the commercial one.

No. Russian has no articles.

So Это семейный бизнес can correspond to English with a or the, depending on context:

• It is a family business.
• This is the family business.

Usually, without special context, learners will most naturally understand it as a family business.

A little, yes, but the original order is the most neutral and natural.

• Это семейный бизнес. = neutral, standard
• Семейный бизнес. = possible as a short label, heading, or very compressed statement
• Этот бизнес семейный. = This business is family-run; different structure
• Бизнес семейный. = possible, but sounds contrastive or stylistically marked

So for a normal textbook sentence, Это семейный бизнес is the best form to learn first.

The stress is:

• Э́то
• семе́йный
• би́знес

A rough English-style guide:

• EH-tuh si-MEY-nyy BEEZ-nyes

A few notes:

• э́ in это is stressed
• ей in семейный sounds like ay in say
• ый at the end of семейный does not exist in English; learners often approximate it as -yy
• бизнес is stressed on the first syllable: БИЗ-

Yes, depending on style and nuance.

Some common alternatives are:

• Это семейное дело.
• Это семейное предприятие.

Differences:

• бизнес = modern, common, direct
• дело = can mean business/venture, but also matter/affair
• предприятие = more formal, closer to enterprise

So Это семейный бизнес is a very natural modern way to say it.