המורה כתבה על הלוח שתי מילים: אחת בזכר ואחת בנקבה.

Yes. The noun מורה can mean either male teacher or female teacher. The verb tells you which one is meant here.

• כתבה = she wrote
• כתב = he wrote

So המורה כתבה tells us the teacher is female.

Because על has more than one meaning in Hebrew, just like on/about can vary by context in English.

In this sentence:

• כתבה על הלוח = wrote on the board

But in another sentence:

• כתבה על הספר could mean wrote about the book

So the context and the noun after על determine the meaning.

The prefix ה־ means the.

• לוח = a board / board
• הלוח = the board

In a classroom situation, it is natural to refer to the specific board everyone knows, so Hebrew uses הלוח.

Because Hebrew uses a special form of 2 before a noun.

For the number two:

• שניים / שתיים = used when the number stands alone
• שני / שתי = used directly before a noun

Since מילה is a feminine noun, you use:

• שתי מילים = two words

Compare:

• שתיים = two
• יש לי שתיים = I have two
• שתי מילים = two words

Because the singular noun is מילה, and מילה is feminine.

You can often tell because many feminine nouns end in ־ה, though that is not the only pattern.

So:

• מילה = feminine singular
• מילים = feminine plural

That means the number must agree with it:

• שתי מילים = correct
• שני מילים = incorrect

Because in Hebrew, after שני / שתי, the noun is normally plural.

So Hebrew says:

• שתי מילים = literally two words

Not:

• שתי מילה

This is normal Hebrew number agreement.

Because את usually marks a definite direct object, and שתי מילים here is not definite.

• שתי מילים = two words (indefinite)
• את שתי המילים = the two words (definite)

So in this sentence, no את is needed.

Because אחת refers to an implied מילה (word), and מילה is feminine.

So the structure is really:

• אחת [מילה] בזכר
• ואחת [מילה] בנקבה

Even though the sentence talks about masculine and feminine forms, the word one is agreeing with the hidden noun מילה, not with זכר or נקבה.

Because Hebrew often leaves out a repeated noun when it is obvious from context.

After שתי מילים, the listener already knows we are talking about words. So:

• אחת בזכר = one [word] in the masculine
• ואחת בנקבה = and one [word] in the feminine

English does this too sometimes, as in one red and one blue.

They mean in the masculine and in the feminine, referring to grammatical gender.

• זכר = masculine
• נקבה = feminine
• בזכר = in masculine form
• בנקבה = in feminine form

Here the teacher wrote two words, and one of them was in a masculine form while the other was in a feminine form.

Not necessarily. In this sentence they refer to grammatical gender, not to whether someone or something is biologically male or female.

Hebrew nouns are assigned grammatical gender:

• some are masculine
• some are feminine

That is a property of the word as part of the language. Sometimes it matches real-world sex, and sometimes it does not.

Yes, this word order is completely natural.

• המורה כתבה על הלוח שתי מילים

Hebrew is somewhat flexible, and you could also say:

• המורה כתבה שתי מילים על הלוח

Both are natural. The given sentence puts על הלוח earlier, which slightly emphasizes where the writing happened.

The colon introduces an explanation or breakdown of the two words.

First the sentence says:

• the teacher wrote two words

Then the part after the colon specifies them by type:

• one in the masculine and one in the feminine

So the colon works much like it does in English: it introduces clarification or a list-like explanation.