Learn Hardcore Latin: In die laboramus et studemus, in nocte cum amicis sedemus et amicitiam sentimus. - In the day we work and study, in the night we sit with friends and feel friendship.

So the subject we (Latin nōs) is understood from the verb endings and does not need to be written. You could add nōs for emphasis, but it is not required:
Nōs in diē labōrāmus et studēmus… would mean something like We (as opposed to others) work and study in the day…

What tense are the verbs laboramus, studemus, sedemus, sentimus, and how can I translate them into English?

All four verbs are in the present tense, indicative mood, active voice, first person plural.

laborāmus – we work / we are working
studēmus – we study / we are studying
sēdēmus – we sit / we are sitting
sentīmus – we feel / we are feeling

Latin present tense can usually be translated in three natural English ways:

we work (simple present)
we are working (present continuous)
we do work (emphatic present)

Context decides which sounds best. Here, we work and study in the day, we sit with friends and feel friendship at night is a natural rendering.

Why does in take die and nocte instead of diem or noctem?

The preposition in can take either the ablative or the accusative case, depending on the meaning:

in
- ablative → rest, position, or time: in, on, at, during
in
- accusative → motion toward: into, onto, against

In this sentence, we are not moving into the day or into the night; we are saying what happens during those times. So we use the ablative:

in diē – in the day / during the day (ablative of time when)
in nocte – in the night / at night (ablative of time when)

If you used the accusative (in diem, in noctem), it would suggest movement into a day or into a night, which is not what is meant here.

Could you leave out in and just say diē laborāmus, nocte sedēmus?

Yes. Latin very often uses the ablative alone for time when without a preposition, especially with words like diē (by day) and nocte (by night).

diē laborāmus – we work by day / in the daytime
nocte cum amīcīs sedēmus – at night we sit with friends

So both are possible:

In diē laborāmus…
Diē laborāmus…

The version without in is actually more classical and a bit more concise; with in is a bit more explicit and is also found, especially in later Latin.

What cases are diē and nocte, and what is their grammatical role?

Both diē and nocte are in the ablative singular and are used as ablatives of time when.

diē – ablative singular of diēs, diēī (day)
nocte – ablative singular of nox, noctis (night)

Their function is to answer when?

(In) diē laborāmus et studēmus – we work and study in the day / during the day
(In) nocte… sedēmus et amīcitiam sentīmus – at night we sit… and feel friendship

So they are not subjects or objects; they are adverbial expressions of time.

Why is it cum amicis and not another case?

The preposition cum (with) always takes the ablative case.

amīcī – friends (nominative plural, subject)
amīcōs – friends (accusative plural, direct object)
amīcīs – friends (ablative plural, here: after cum)

So cum amīcīs literally means with friends.

Its grammatical role is to show accompaniment: we are not just sitting; we are sitting with friends. It does not change the subject of the verb; the subject is still we, from the verb ending -mus.

Who is the subject of sedemus – we or the friends?

The subject is still we.

The verb sedēmus has the ending -mus, which marks first person plural → we sit.
cum amīcīs is an ablative phrase of accompaniment → with friends, not the subject.

So the meaning is:

In nocte cum amīcīs sedēmus – At night we sit with friends,
not
At night the friends sit…

If the friends were the subject, the verb ending would match them, for example amīcī sedent – the friends sit.

Why is amicitiam in a different form from amicis, and what is its function?

amīcitiam and amīcīs come from different nouns and serve different grammatical roles:

amīcitia, amīcitiae (f.) – friendship
- amīcitiam here is accusative singular → direct object of sentīmus.
- We feel friendship.
amīcus, amīcī (m.) – friend
- amīcīs here is ablative plural after cum → with friends.

So in the second clause:

cum amīcīs – with friends (who are with us)
amīcitiam sentīmus – we feel friendship (what we feel)

Friends are the company, friendship is the thing felt.

Why can studemus stand on its own without an object?

The verb studeō, studēre means to be eager for, to apply oneself to, to study. It can be used in several patterns:

With a dative: studeō litterīs – I study letters / literature
With an infinitive: studeō discere – I am eager to learn
Or absolutely, without adding what exactly you study: simply I study.

In this sentence, studemus is used in that general sense:

In diē laborāmus et studēmus – in the day we work and we study (in general)

Latin does not have to specify what you study, just as in English it is natural to say I work and study all day without adding a direct object.

Why is the Latin word order different from English? Could you rearrange it?

Latin word order is much more flexible than English, because the endings (cases, verb endings) show grammar relations.

The original:

In diē laborāmus et studēmus, in nocte cum amīcīs sedēmus et amīcitiam sentīmus.

You could also see, for example:

Diē in labōrāmus et studēmus…
Diē labōrāmus et studēmus, nocte cum amīcīs sedēmus et amīcitiam sentīmus.
Cum amīcīs in nocte sedēmus et amīcitiam sentīmus.

All of these would still mean essentially the same thing, because:

The verbs tell you the subject (-mus → we).
The cases tell you roles: amīcitiam (accusative) is the thing felt; amīcīs (ablative) is the company; diē, nocte (ablative) are times.

Latin often likes to put important or contrasting words toward the beginning or end of a clause; there is no single fixed order like English Subject–Verb–Object.

How should I think about translating in die and in nocte into natural English: in, on, at, or during?

The Latin ablative with in here expresses time when, and it is broader than any single English preposition. Depending on context, all of these can be reasonable:

in diē → in the day, in the daytime, during the day
in nocte → in the night, at night, during the night

The most idiomatic English here is likely:

During the day we work and study, at night we sit with friends and feel friendship.

So when you see in + ablative used for time, think in your head: at / in / during that time, and then choose the English preposition that sounds most natural in context.