Learn Hardcore Russian: Сейчас мой брат свободен вечером, а завтра он будет занят до поздней ночи. - Right now my brother is free in the evening, and tomorrow he will be busy until late at night.

In this sentence, Сейчас мой брат свободен вечером..., сейчас has the meaning “at the present time / these days”. It’s about his current schedule, not a contrast with the past, so сейчас is more natural than теперь here.
(You could say Теперь мой брат свободен вечером, but that would imply something changed compared to earlier.)

Why свободен and not свободный in мой брат свободен?

Russian has short-form adjectives (свободен, занята, рады, etc.) and long-form adjectives (свободный, занятый, радостный, etc.).

Short form is usually used as a predicate, to describe a temporary state / condition:
- Мой брат свободен. – My brother is free (right now / at the moment).
Long form is usually attributive (before a noun) or about a more permanent characteristic:
- свободный человек – a free person
- У него свободный график. – He has a flexible schedule.

So:

Мой брат свободен вечером. – He is free in the evenings (currently, as a state).
Мой брат свободный. – He is a free person (context-dependent: not in a relationship, not in prison, free-minded, etc.).
That would sound strange here if you just mean “he’s not busy.”

Where is the verb “to be”? Why isn’t it есть: Сейчас мой брат есть свободен?

In Russian, the verb быть (to be) in the present tense is usually omitted in statements like this.

Full theoretical form: Сейчас мой брат (есть) свободен вечером.
Natural Russian: Сейчас мой брат свободен вечером.

So:

Он свободен. = “He is free.” (no есть)
Он будет свободен. = “He will be free.” (future of быть is used)
Он был свободен. = “He was free.” (past of быть is used)

You only see есть in other meanings or in emphasis, but not in simple “X is Y” present-tense sentences.

What does Сейчас мой брат свободен вечером mean in terms of time? “Right now” or “these days”?

Here сейчас plus вечером together give the idea of a current, ongoing situation:

Сейчас мой брат свободен вечером ≈
“At the moment (these days), my brother is free in the evenings.”

So it’s not “he is free this particular evening right now,” but rather:

right now in his life / schedule, evenings are free.

If you wanted to say “this evening he is free”, you’d say:

Сегодня вечером мой брат свободен. – My brother is free this evening.

What form is вечером, and why does it mean “in the evening”?

Вечером is the instrumental singular form of вечер (evening), but in practice it functions as an adverbial form of time:

утром – in the morning
днём – in the daytime / during the day
вечером – in the evening
ночью – at night

So:

Мой брат свободен вечером. – My brother is free in the evening / in the evenings (context decides whether it’s one evening or generally).

Other related forms:

вечер – evening (basic noun form)
по вечерам / вечерами – in the evenings (regularly, on multiple evenings)

Why is а used instead of и before завтра?

Both а and и can be translated as “and”, but they have different functions:

и = simple “and”, adding similar information.
а = “and / but / whereas”, shows contrast or difference.

In the sentence:

Сейчас мой брат свободен вечером, а завтра он будет занят до поздней ночи.

А highlights the contrast:

Now he is free in the evening, but tomorrow he will be busy until late at night.

Using и here would sound weaker and less natural, because the point is to contrast today’s freedom with tomorrow’s busy schedule.

Why do we say завтра он будет занят, not just завтра он занят?

Both are grammatically possible, but there is a nuance:

Завтра он будет занят. – clear future tense, “He will be busy tomorrow.”
Завтра он занят. – present form referring to a scheduled future:
“He’s busy tomorrow” (like English “I’m busy tomorrow”).

In this sentence, будет занят is used to make the future explicit and symmetric with the present clause:

Сейчас — свободен,
завтра — будет занят.

It emphasizes the change from now to tomorrow.

What exactly is занят? Is it an adjective or a participle?

Занят is the short-form of the adjective / participle занятый (“busy, occupied”). It functions like a short-form adjective used in the predicate:

masculine: занят – Он занят. (He is busy.)
feminine: занята – Она занята. (She is busy.)
neuter: занято – Дело занято. (The matter is occupied/busy.)
plural: заняты – Мы заняты. (We are busy.)

In он будет занят, it describes his state (“he will be busy”), not an active action he’s doing.

What case is поздней ночи in до поздней ночи, and why?

Ночи is in the genitive singular of ночь (night), and поздней is the genitive singular feminine form of the adjective поздний (late).

The preposition до (до + Genitive) always requires the genitive case and means “up to / until”:

до трёх часов – until three o’clock
до завтра – until tomorrow
до вечера – until evening
до поздней ночи – until late at night

So до поздней ночи literally = “up to (a) late night,” idiomatically: “until late at night.”

Could we say до поздно instead of до поздней ночи?

You can say до поздна (note the form!) in Russian, and it means “until late” in a general sense:

Он работает до поздна. – He works until late.

However:

до поздней ночи specifies the part of the day (night) and sounds more vivid and precise, like English “until late into the night.”
до поздна is more neutral and less specific.

In your sentence, до поздней ночи emphasizes being busy deep into the night, not just “late.”

What exactly does поздней agree with, and how is it formed?

Поздней is the genitive singular feminine form of the adjective поздний (late), and it agrees with ночи:

поздняя ночь – late night (Nom. sg. fem.)
поздней ночи – of (a) late night / until late night (Gen. sg. fem.)

Pattern:

Nom. fem.: поздняя ночь
Gen. fem.: поздней ночи

Because до requires genitive, both the noun and adjective take their genitive forms: до поздней ночи.

Can we change the word order, for example: Сейчас вечером мой брат свободен?

Yes, Russian word order is flexible, but it changes emphasis:

Сейчас мой брат свободен вечером.
Neutral: “Right now (these days), my brother is free in the evenings.”
Сейчас вечером мой брат свободен.
Sounds a bit odd; сейчас вечером together is usually understood as “this evening right now,” which clashes with сейчас. Better is:
- Сегодня вечером мой брат свободен. – My brother is free this evening.
Мой брат сейчас свободен вечером.
Still understandable, but the focus shifts slightly to мой брат as the topic; the meaning stays similar.

The original order is the most natural and clear for the intended meaning (current general evening schedule).

Why not say будет занятым instead of будет занят?

Будет занят (short form) is the normal, idiomatic way to say “will be busy.”

Он будет занят. – He will be busy. (natural)
Он будет занятым. – Grammatically possible but sounds unnatural / overly formal or wrong in everyday speech.

With быть + predicate describing a state, Russian strongly prefers the short form:

Он был рад. – He was glad. (not радостным here)
Она будет готова. – She will be ready.
Мы были заняты. – We were busy.

Use занятый (long form) before nouns or in some more complex constructions:

занятый человек – a busy person
But not будет занятым in this simple sense.

How would you say the feminine or plural versions of these phrases?

Using the same pattern with short-form adjectives:

Masculine:
- Он свободен вечером. – He is free in the evening.
- Он будет занят до поздней ночи. – He will be busy until late at night.
Feminine:
- Она свободна вечером. – She is free in the evening.
- Она будет занята до поздней ночи. – She will be busy until late at night.
Plural:
- Они свободны вечером. – They are free in the evening.
- Они будут заняты до поздней ночи. – They will be busy until late at night.

Notice the short-form adjective changes for gender/number:

свободен / свободна / свободно / свободны
занят / занята / занято / заняты