Pystytämme teltan yötä varten.

Pystytämme is the we form of pystyttää, which means to erect, set up, pitch.

So:

• pystyttää = to put up / set up
• pystytämme = we set up / we are setting up / we will set up

This is easy to confuse with pystyä, which means to be able to.

Compare:

• Me pystymme = we can
• Me pystytämme = we set up

They look similar, but they are different verbs.

Because teltan is the object form used here.

The basic dictionary form is:

• teltta = tent

In this sentence, the tent is the thing being completely set up, so Finnish uses the total object. In the singular, that often looks like the genitive form:

• teltta → teltan

So:

• Pystytämme teltan = We pitch the tent

The idea is that the action is directed at the whole tent as a completed result.

A useful contrast:

• Pystytämme teltan = we put up the whole tent
• Pystytämme telttaa = we are putting up a tent / the tent, with a more ongoing, incomplete, or indefinite feel

Because varten requires the noun before it to be in the partitive case.

The basic form is:

• yö = night

The partitive singular is:

• yötä

So:

• yötä varten = for the night

This does not mean part of a night here. It is simply the form required by the postposition varten.

Varten means something like:

• for
• for the purpose of
• in preparation for

It comes after the noun because it is a postposition, not a preposition.

English usually uses prepositions:

• for the night

Finnish often uses postpositions:

• yötä varten
• literally: night for

So the structure is normal Finnish:

• noun in partitive + varten

Examples:

• matkaa varten = for the trip
• vieraita varten = for guests
• talvea varten = for the winter

Grammatically, it is present tense.

But Finnish often uses the present tense to talk about the future when the context makes it clear. So this sentence can mean:

• We are pitching the tent for the night
• We will pitch the tent for the night

Finnish does not have a separate future tense like English does. Context tells you whether it is present or future.

Because the verb ending already tells you the subject.

In pystytämme, the ending -mme means we.

So Finnish does not need the pronoun:

• Pystytämme teltan yötä varten. = We pitch the tent for the night.

You can add me:

• Me pystytämme teltan yötä varten.

But then it usually sounds more emphatic, contrastive, or explicit, like:

• We are the ones pitching the tent

So leaving out me is completely normal.

Yes, you could, and it would sound very natural.

• yöksi = for the night / for overnight

So:

• Pystytämme teltan yöksi
  = We pitch the tent for the night

The difference is mainly nuance:

• yötä varten emphasizes purpose or preparation for the night
• yöksi emphasizes the idea of for the duration of the night / overnight

In many real situations, the meanings are very close.

Yes, Finnish word order is fairly flexible, though the basic order here is very natural:

• Pystytämme teltan yötä varten.

This is a neutral order: verb + object + purpose phrase.

You could move things around for emphasis. For example:

• Yötä varten pystytämme teltan.

This puts more focus on for the night.

Even when the order changes, the case endings help show what each word is doing, so the meaning usually stays clear.

Because Finnish verbs change their stem when they are conjugated.

The dictionary form is:

• pystyttää

But the present-tense forms are built from a conjugated stem, not by just attaching endings unchanged.

For example:

• minä pystytän = I set up
• me pystytämme = we set up

So pystytämme is just the normal conjugated we form of pystyttää.

For a learner, the most useful thing is to remember the pattern through actual forms rather than trying to keep the infinitive unchanged in every form.