Мы жили там полгода.

Жили is the past tense plural form of жить (to live).

It is also imperfective. That matters because Russian uses the imperfective to describe:

• an ongoing state
• a repeated/habitual action
• a situation viewed as a process or duration

So Мы жили там полгода presents the living there as a state that lasted for half a year.

In the Russian past tense, verbs agree with the subject in gender and number.

Because мы means we, the verb must be plural, so you get:

• я жил / я жила
• ты жил / ты жила
• он жил
• она жила
• оно жило
• мы жили
• вы жили
• они жили

So the -и shows plural past.

Yes. Russian often drops subject pronouns when they are clear from the verb form.

So Жили там полгода can be understood as We lived there for half a year, depending on context.

Including мы can make the subject clearer or slightly more emphatic:

• Мы жили там полгода = We lived there for half a year.

Because там means there in the sense of location, while туда means to there / there in the sense of direction.

Here the sentence describes where the living happened, not movement toward a place.

• там = in that place
• туда = to that place

So:

• Мы жили там = We lived there.
• Мы поехали туда = We went there.

Because полгода is a fixed Russian word meaning half a year.

It is built from:

• пол- = half
• года = form of год (year)

Russian often writes this kind of expression as one word:

• полгода = half a year
• полчаса = half an hour

This is a very common and natural way to express time length.

After пол- (half), Russian normally uses the genitive singular form of the noun.

The noun is год (year), and its genitive singular is года.

So:

• год = a year
• полгода = half of a year

Even though English learners may want to think of the whole thing as for half a year, internally Russian is using the idea half of a year.

Russian often expresses duration of time with no preposition at all.

So where English says:

• for half a year
• for two days
• for a long time

Russian can simply say:

• полгода
• два дня
• долго

That means полгода is functioning as an adverbial expression of duration: it tells you how long the action/state lasted.

This is an aspect question.

• жили = imperfective
• прожили = perfective

Both can be translated similarly, but the nuance changes:

• Мы жили там полгода focuses on the state/duration of living there.
• Мы прожили там полгода presents that half-year as a completed whole, almost like we spent half a year living there.

In many contexts, both are possible, but жили is the more neutral descriptive version.

Yes. Russian word order is flexible, and changing it usually changes focus rather than basic meaning.

Possible versions include:

• Мы жили там полгода.
• Мы там жили полгода.
• Полгода мы жили там.
• Там мы жили полгода.

All are grammatical, but they highlight different pieces of information.

For example:

• Полгода мы жили там puts emphasis on the length of time.
• Там мы жили полгода puts more emphasis on the place.

Often, yes.

Both mean roughly the same thing, but:

• полгода = half a year
• шесть месяцев = six months

Полгода is very natural in everyday Russian when you want a simple, compact time expression.
Шесть месяцев can sound a bit more precise, formal, or measured, depending on context.

So in ordinary speech, Мы жили там полгода sounds very natural.

A simple pronunciation guide:

• Мы = mɨ
• жили = ZHEE-li with the stress on the first syllable
• там = tam
• полгода = POL-ga-da with the stress on пол

So the stress pattern is:

МЫ ЖИ-ли ТАМ ПОЛ-го-да

More accurately:

• жи́ли
• по́лгода

Most naturally, it means one period of time: they lived there for half a year.

Because жить is imperfective, it can in some contexts describe habitual or repeated situations, but with полгода here, the default reading is a single stretch of time lasting half a year.

So learners will usually understand it as:

• one continuous period of living there lasting half a year.