Завтра будет среда, шестое марта, и у нас будет короткий урок.

Russian does not need a dummy subject like English it in expressions about time, dates, or weather.

So:

• Завтра будет среда = literally Tomorrow will be Wednesday
• natural English = Tomorrow it will be Wednesday

In Russian, будет среда is a normal impersonal-style statement about the calendar.

Because шестое does not agree with среда.

In Russian dates, the day number is usually understood as an omitted neuter noun: число (date / number). So шестое марта is really:

• шестое (число) марта = the sixth of March

That is why шестое is neuter singular. It is not describing среда; it is giving a separate piece of information: the calendar date.

Because in Russian dates, the month is normally put in the genitive case after the day number.

So:

• шестое марта = the sixth of March
• первое апреля = the first of April
• двадцать пятое декабря = the twenty-fifth of December

So марта is the regular form used in dates.

Because the sentence contains two separate future ideas:

• Завтра будет среда = Tomorrow will be Wednesday
• у нас будет короткий урок = we will have a short lesson

In both parts, будет is the future form of быть (to be). Russian often repeats the verb where English might also repeat will be / will have.

Literally, у нас means by us / at our place / with us. But very often Russian uses у + genitive to express something like have.

So:

• у нас будет короткий урок = literally at us there will be a short lesson
• natural English = we will have a short lesson

This is much more natural in Russian than something like мы имеем короткий урок, which would sound unnatural in everyday speech.

Because adjectives must agree with the noun they describe.

Here, урок is:

• masculine
• singular
• nominative

So the adjective must also be masculine singular nominative:

• короткий урок

If the noun were neuter, the adjective would change:

• короткое занятие = a short class/session

Russian has no articles like a or the.

So короткий урок can mean:

• a short lesson
• the short lesson

The exact meaning depends on context. In this sentence, English naturally uses a short lesson, but Russian does not need a separate word for that.

Yes, often you can, especially in conversation.

For example:

• Завтра среда, шестое марта, и у нас короткий урок.

Russian often uses the present tense to talk about scheduled or expected future events. But the version with будет is also completely correct and makes the future meaning very explicit.

Because in Russian, names of days of the week and months are normally not capitalized.

So Russian writes:

• среда
• март
• марта

where English writes:

• Wednesday
• March

They are capitalized only at the beginning of a sentence or as part of a title/name.

Because шестое марта is giving an additional clarification of среда. It tells you exactly which Wednesday is meant.

So the structure is roughly:

• Tomorrow will be Wednesday, the sixth of March...

The date works like an explanatory apposition, and the commas mark that extra identifying information.

Here it means short in duration, so brief.

For lessons, classes, meetings, and similar events, короткий is a normal word for something that does not last long. So короткий урок means a lesson with less time than usual, not a physically small lesson.