Вона зважила кілограм моркви.

Because it is past tense feminine singular.

In Ukrainian past tense, the verb agrees with the subject in gender and number:

• зважив = he weighed
• зважила = she weighed
• зважило = it weighed
• зважили = they weighed

So with Вона (she), the form must be зважила.

зважила is perfective, so it presents the action as completed: she finished weighing it.

That fits a sentence about one whole finished action.

A rough contrast:

• Вона зважила кілограм моркви. = she weighed out / weighed a kilogram of carrots (completed action)
• Вона зважувала кілограм моркви. = she was weighing / used to weigh a kilogram of carrots (process, repetition, or background action)

So the prefix in зважила helps give the sense of a single completed result.

Here кілограм is in the accusative singular, because it is the direct object of зважила.

But for a masculine inanimate noun like кілограм, the accusative singular looks the same as the nominative singular. So:

• nominative: кілограм
• accusative: кілограм

That is why it looks unchanged.

Because after a measure word like кілограм, Ukrainian normally puts the thing being measured into the genitive.

So:

• кілограм моркви = a kilogram of carrots / carrot

The main noun is кілограм, and моркви depends on it.

If you said моркву, that would be accusative, which would not fit this structure.

There is no separate word for of here.

Ukrainian usually expresses that relationship with the genitive case:

• кілограм моркви = literally kilogram carrots-GEN

So the idea of of is carried by the ending of моркви, not by a separate word.

Because Ukrainian often treats foods and produce sold or measured by weight as a mass noun.

So моркви here is genitive singular of морква, and the phrase means something like:

• a kilogram of carrot
• which English naturally translates as a kilogram of carrots

This is very normal in Ukrainian. English prefers the plural here, but Ukrainian often prefers the singular mass-style form.

Кілограм by itself is enough.

In this kind of phrase, кілограм naturally means a/one kilogram unless you specify another number.

So both are possible:

• Вона зважила кілограм моркви.
• Вона зважила один кілограм моркви.

The version without один is more neutral and natural unless you specifically want to stress one kilogram.

Yes. Ukrainian word order is fairly flexible because the case endings show the grammatical relationships.

The neutral order here is:

• Вона зважила кілограм моркви.

But you could also say:

• Кілограм моркви вона зважила.
  This emphasizes a kilogram of carrots.
• Моркви вона зважила кілограм.
  This is possible, but more marked and context-dependent.

So the original version is the most straightforward, but other orders are possible for emphasis.

Yes, often you can, if the context makes the subject clear:

• Зважила кілограм моркви.

However, there is an important detail: Ukrainian past tense shows gender and number, but not person very clearly in the same way English does. So зважила could mean:

• she weighed
• I weighed (if the speaker is female)
• you weighed (speaking to one female, depending on context)

So вона is useful when you want to make it explicitly she.

It can also have a figurative meaning like considered or weighed carefully in other contexts.

For example, in another sentence, зважити can mean to weigh options or to consider something.

But in Вона зважила кілограм моркви, the meaning is clearly the literal one, because кілограм моркви makes the physical weighing obvious.