הוא כבר ארז את המזוודה שלו, אבל אני עוד לא ארזתי את שלי.

Everyday Hebrew usually does not use a separate tense like the English present perfect.

So:

• הוא כבר ארז can mean he packed or he has already packed
• אני עוד לא ארזתי can mean I didn’t pack yet or more naturally I haven’t packed yet

Hebrew normally uses the past tense form and lets context do the rest.

Also, Hebrew does not need an extra helping verb like have or do here.

כבר means already.

So:

• הוא כבר ארז = He already packed / He has already packed

It often appears before the verb, as it does here.

עוד לא is a very common combination meaning not yet or still not.

So:

• אני עוד לא ארזתי = I haven’t packed yet / I still haven’t packed

Word by word, עוד often has the idea of still / more / yet, and לא means not. But as a phrase, it is best learned as not yet.

Hebrew does not use extra helper words like English does.

English says:

• I haven’t packed
• I didn’t pack
• I don’t pack

Hebrew usually just uses לא with the verb form itself.

Here:

• לא ארזתי = I didn’t pack / I haven’t packed

Then עוד adds the idea of yet:

• עוד לא ארזתי = I haven’t packed yet

In Hebrew, את marks a definite direct object.

Here, המזוודה שלו is definite because it means his suitcase, a specific suitcase.

So:

• ארז את המזוודה שלו = packed his suitcase

You usually use את before a direct object that is definite, such as:

• a noun with ה־
• a name
• a pronoun
• a possessed noun like המזוודה שלו

Because שלי here stands for an understood noun: my suitcase.

So:

• את שלי = mine / my one
• in full: את המזוודה שלי

Since the missing noun is still a definite direct object, את is still needed.

That is why אני עוד לא ארזתי את שלי is natural, but אני עוד לא ארזתי שלי is not standard.

They are possessive forms built with של:

• שלו = his / literally of him
• שלי = my / mine / literally of me

So:

• המזוודה שלו = his suitcase
• שלי = mine / my one

A useful point: these forms depend on the owner, not on the gender of the thing owned.

For example:

• שלו = belonging to him
• שלה = belonging to her
• שלי = belonging to me regardless of whether the speaker is male or female

Because the suitcase is specific/definite.

In Hebrew, when you say his suitcase, the noun is usually treated as definite, so you normally say:

• המזוודה שלו

This is the common everyday pattern:

• הספר שלי = my book
• הבית שלהם = their house

So Hebrew often looks like the suitcase of him, even though natural English is just his suitcase.

Because Hebrew, like English, often leaves out a repeated noun when it is obvious.

So instead of repeating המזוודה again, Hebrew just says:

• את שלי = mine / my one

The full version would be:

• אבל אני עוד לא ארזתי את המזוודה שלי

But the shorter version is more natural because the noun is already understood.

Because Hebrew verbs change according to the subject.

Here the verb is from the root א־ר־ז, connected with packing.

• ארז = he packed
• ארזתי = I packed

So the endings tell you who did the action.

This is one reason Hebrew often does not need subject pronouns.

Yes, it could.

Hebrew often omits subject pronouns because the verb already shows the person:

• ארזתי already means I packed

So this is also natural:

• אבל עוד לא ארזתי את שלי

Including אני adds contrast or emphasis:

• He already packed his suitcase, but I haven’t packed mine yet.

That contrast is probably why אני appears here.

Because the verb agrees with the subject, not with the object.

The subject here is:

• הוא = he

So the verb is masculine singular:

• ארז

The object מזוודה being feminine does not affect the verb.

Compare:

• הוא כבר ארז את המזוודה שלו = He already packed his suitcase
• היא כבר ארזה את המזוודה שלה = She already packed her suitcase

The change from ארז to ארזה happens because the subject changes from he to she.

This word order is very natural and neutral:

• הוא כבר ארז את המזוודה שלו
• אבל אני עוד לא ארזתי את שלי

Hebrew word order is somewhat flexible, but moving things around usually changes the emphasis.

For example, you could emphasize the object:

• את המזוודה שלו הוא כבר ארז

That still means the same basic thing, but it sounds more marked or emphatic.

So the sentence you were given is the most straightforward everyday order.

Yes. Hebrew does have more compact possessive forms in some cases.

So his suitcase could also appear as:

• מזוודתו

But that style is usually more formal, literary, or less common in everyday conversation.

In ordinary modern speech, most people would prefer:

• המזוודה שלו

So the version in your sentence is the normal spoken style.