הוא מזמין דג עם אורז, אבל אני רוצה בשר.

Yes. The verb להזמין can mean both to order and to invite.

In this sentence, the context makes to order the natural meaning, because the object is food: דג עם אורז.

Also, Hebrew present tense does not separate he orders and he is ordering the way English does. So הוא מזמין can mean either, depending on context.

Because את is used only before a definite direct object.

Here, דג and בשר are indefinite:

• הוא מזמין דג = he is ordering fish / a fish dish
• אני רוצה בשר = I want meat / some meat

If the nouns were definite, you would use את:

• הוא מזמין את הדג = he is ordering the fish
• אני רוצה את הבשר = I want the meat

Hebrew usually does not have a separate word for the indefinite article a/an.

So:

• דג can mean a fish or fish
• בשר can mean meat or some meat
• אורז can mean rice or some rice

If you want to make a noun definite, Hebrew normally adds ה־:

• דג = fish / a fish
• הדג = the fish

עם means with, so עם אורז means with rice.

In this sentence, it most naturally describes the food order:

• דג עם אורז = fish with rice

So the idea is that he is ordering a fish dish that comes with rice, or fish served together with rice.

Sometimes Hebrew can omit subject pronouns, but in the present tense they are often kept because the verb form usually does not clearly show the person.

For example:

• מזמין tells you masculine singular, but not by itself whether it means he is ordering, you are ordering (to a male), or something else from context.
• רוצה also does not clearly mark person on its own.

So in a sentence like this, הוא and אני help make the meaning clear.

Because in unpointed Hebrew, the masculine and feminine singular forms of this verb are spelled the same:

• masculine: רוצה = pronounced rotze
• feminine: רוצה = pronounced rotza

So:

• a male speaker says אני רוצה = ani rotze
• a female speaker says אני רוצה = ani rotza

Same spelling, different pronunciation.

If he became she, you would say:

היא מזמינה דג עם אורז, אבל אני רוצה בשר.

The change is:

• הוא מזמין = he is ordering
• היא מזמינה = she is ordering

If the speaker אני is female, רוצה stays spelled the same, but is pronounced rotza instead of rotze.

Because Modern Hebrew often uses the basic word order Subject–Verb–Object, especially in everyday speech.

So:

• הוא = subject
• מזמין = verb
• דג עם אורז = object phrase

Then the second clause works the same way:

• אני = subject
• רוצה = verb
• בשר = object

Hebrew can use other word orders for emphasis or style, but this sentence is very normal and straightforward.

It can mean either in different contexts, but here it clearly means fish as food.

That is because of the restaurant-like context:

• מזמין = ordering
• עם אורז = with rice

So a learner should understand דג here as a fish dish, not a live fish.

אבל means but. It connects two contrasting ideas:

• הוא מזמין דג עם אורז
• אבל אני רוצה בשר

So the contrast is:

• he is ordering fish with rice
• but I want meat

It works very much like English but in this sentence.