אל תשימי את השקית ליד הדלת; תשימי אותה במטבח.

Because of תשימי.

That verb form is second person feminine singular. Hebrew marks gender in you forms, unlike English.

So here:

• תשימי = you (feminine singular) will put / put
• If speaking to a man, you would use תשים
• If speaking to more than one person, you would use תשימו

So the sentence is directed at one female addressee.

Hebrew usually does not need a separate word for you, because the verb already tells you who the subject is.

In תשימי, the ending already shows:

• second person
• feminine
• singular

So Hebrew can leave out את meaning you unless there is special emphasis.

Important: the את in את השקית is not the pronoun you. Here it is the direct object marker, which is a different word even though it is spelled the same way.

Because אל is used for a negative command:

• אל תשימי = don’t put
• לא is usually for negating statements, not commands

Compare:

• אל תשימי את השקית ליד הדלת = Don’t put the bag by the door
• את לא שמה את השקית ליד הדלת = You are not putting the bag by the door

So for prohibitions like don’t do X, Hebrew normally uses אל + future-form verb.

Yes, תשימי is formally a future form, but in Hebrew future forms are also often used for commands, especially in everyday language.

So in this sentence:

• אל תשימי = negative command: don’t put
• תשימי אותה במטבח = command/request: put it in the kitchen

In the second clause, Hebrew could also use the true imperative:

• שימי אותה במטבח

That would also be completely natural.

So:

• שימי = imperative
• תשימי = future form, but often used like a command in speech

Here את is the direct object marker.

Hebrew puts את before a definite direct object. Since השקית means the bag and is definite because of ה־, it takes את:

• את השקית = the bag as a direct object

This את usually is not translated into English.

Compare:

• תשימי את השקית = put the bag
• תשימי שקית = put a bag / put bag
  No את, because the object is indefinite

Because Hebrew nouns have grammatical gender.

שקית is a feminine noun, so when you refer back to it with it, Hebrew uses the feminine object pronoun:

• אותה = her / it for a feminine singular noun

So:

• השקית ... תשימי אותה = the bag ... put it

If the noun were masculine, Hebrew would use:

• אותו = masculine singular him / it

English says it for most objects, but Hebrew still keeps track of whether the noun is masculine or feminine.

Sometimes yes, if the meaning is obvious from context.

For example, in conversation someone might say:

• תשימי במטבח

and the listener understands what is being put there.

But in this sentence, אותה makes the reference very clear and sounds more complete:

• אל תשימי את השקית ליד הדלת; תשימי אותה במטבח

So omission is possible in some contexts, but keeping אותה is very natural here.

Because these are two different kinds of prepositions.

1. ב־ is a prefixed preposition

It attaches directly to the noun:

• במטבח = in the kitchen

When ב־ combines with the definite article ה־, they merge in pronunciation.

2. ליד is a separate word

It stands on its own:

• ליד הדלת = next to the door

So you get:

• במטבח = in the kitchen
• ליד הדלת = next to the door

That difference is completely normal in Hebrew.

The word order here is the most neutral and common one for a command:

• אל תשימי את השקית ליד הדלת
• תשימי אותה במטבח

That is basically:

• verb + object + place

But Hebrew word order is somewhat flexible. You can move parts around for emphasis.

For example:

• את השקית אל תשימי ליד הדלת
  This emphasizes the bag
• במטבח תשימי אותה
  This emphasizes in the kitchen

So the original order is the standard, unmarked one, but other orders are possible when the speaker wants a different emphasis.

Yes. That would be very natural.

So these are both possible:

• אל תשימי את השקית ליד הדלת; תשימי אותה במטבח
• אל תשימי את השקית ליד הדלת; שימי אותה במטבח

The difference is mainly stylistic:

• שימי = straightforward imperative
• תשימי = future form used as a command, often common in everyday speech

Using תשימי in both clauses can also create a nice parallel pattern:

• אל תשימי ...; תשימי ...

So the sentence you were given is correct, but שימי would also be a very natural alternative.