Imperator heredium veterano assignat.

Because it is the thing being assigned. With assignare, the thing assigned is put in the accusative case, and heredium is functioning that way here.

A useful extra point: heredium is a neuter noun, so its nominative and accusative singular forms look the same. Even so, its role is clear because imperator is the subject and veterano is the recipient.

Assignat is present active indicative, third person singular of assignare.

So it means:

• he assigns
• she assigns
• it assigns

In this sentence, with imperator as the subject, it means the commander/emperor assigns.

Latin word order is much freer than English word order because Latin uses case endings to show each word's job in the sentence.

English usually relies heavily on position:

• The commander assigns the estate to the veteran.

Latin can say:

• Imperator heredium veterano assignat.

A very common Latin tendency is to place the verb near the end, but that is not a fixed rule.

Yes. Because the endings show the grammar, Latin can move the words around much more easily than English.

For example, these would keep the same basic meaning:

• Imperator veterano heredium assignat
• Heredium imperator veterano assignat
• Veterano imperator heredium assignat

The main difference would be emphasis, not the core meaning.

That depends on context.

In many Latin texts, imperator means commander or general. In later Roman usage, it can also mean emperor. Grammatically, either is possible here; the surrounding passage would tell you which translation is best.