היא חותכת גזר קטן ושמה אותו ליד היוגורט.

Yes. Both חותכת and שמה are feminine singular forms, so the sentence could often work without היא if the subject is clear from context.

So:

• היא חותכת גזר קטן... = She is cutting a small carrot...
• חותכת גזר קטן... can also mean She is cutting a small carrot... if the context already makes the subject clear

Hebrew often drops subject pronouns, especially in informal speech, but keeping היא can make the sentence clearer or slightly more explicit.

They are both present-tense feminine singular forms, matching the subject היא.

• חותכת = cutting / cuts
• שמה = putting / puts

In Hebrew, the present tense is built from forms that also behave a bit like participles, and they must agree with the subject in gender and number.

Compare:

• הוא חותך ושם = he cuts and puts
• היא חותכת ושמה = she cuts and puts

So the ending and form change because the subject is feminine singular.

It comes from לשים, which means to put / to place, and this verb is somewhat irregular.

Some common forms are:

• לשים = to put
• שם = he puts / putting
• שמה = she puts / putting
• שמתי = I put
• ישים = he will put

So even though שמה may not immediately look like לשים, it is the correct present feminine singular form of that verb.

The ו is the Hebrew word for and, and it is usually attached directly to the next word.

So:

• ושמה = ו + שמה = and she puts

This is very common in Hebrew:

• ואוכל = and I eat
• ובית = and a house / and house
• והילד = and the boy

In Hebrew, adjectives usually come after the noun they describe.

So:

• גזר קטן = a small carrot

not:

• קטן גזר

Also, the adjective must agree with the noun in gender and number.

Here:

• גזר is masculine singular
• so the adjective is קטן = small in masculine singular

If the noun were feminine, the adjective would change too:

• עגבנייה קטנה = a small tomato

Because את is used before a definite direct object, and גזר קטן here is indefinite: a small carrot, not the small carrot.

So:

• היא חותכת גזר קטן = she cuts a small carrot
• היא חותכת את הגזר הקטן = she cuts the small carrot

That is a very important rule in Hebrew: את usually appears only with definite direct objects.

אותו here means it, referring back to גזר.

It is masculine because גזר is a masculine noun in Hebrew. Hebrew pronouns agree with the grammatical gender of the noun, even for things that would just be it in English.

So:

• גזר is masculine → אותו
• if the noun were feminine, you would use אותה

For example:

• היא לוקחת את הצלחת ושמה אותה על השולחן
  The noun צלחת is feminine, so the pronoun is אותה.

Yes, historically and grammatically they are related.

A helpful way to think of אותו is as a built-in direct-object pronoun meaning him / it (masculine singular). In modern Hebrew, learners usually just memorize it as the independent object pronoun:

• אותי = me
• אותך = you
• אותו = him / it
• אותה = her / it

So in this sentence, שמה אותו means she puts it.

In writing without vowel marks, שמה can indeed be ambiguous.

It can be:

• שָׂמָה = she puts
• שָׁמָה = there / to there

But in this sentence, it clearly means she puts, because it follows the conjunction ו and is followed by the object אותו and a location phrase:

• ושמה אותו ליד היוגורט = and she puts it next to the yogurt

So the context makes the meaning clear.

ליד is a preposition meaning next to / beside.

So:

• ליד היוגורט = next to the yogurt

The noun יוגורט has the definite article ה־ attached to it:

• יוגורט = yogurt
• היוגורט = the yogurt

That is how Hebrew usually expresses the: by adding ה־ to the beginning of the noun.

Yes. In Hebrew, if a noun is definite, its adjective also becomes definite.

So compare:

• גזר קטן = a small carrot
• הגזר הקטן = the small carrot

Both the noun and the adjective take ה־ when the whole noun phrase is definite. That is a very common pattern in Hebrew.

Yes. This is a very natural and standard word order:

• היא = subject
• חותכת גזר קטן = verb + object
• ושמה אותו ליד היוגורט = and + verb + object + location

So the sentence follows a common pattern:

subject + verb + object + additional information

Hebrew can be flexible with word order, but this version is straightforward and natural for everyday use.