Nos laeti sumus, et multum laboramus.

Because Latin verb endings already show the subject, nos is not required for basic grammar.

• sumus = we are
• laboramus = we work

So nos is added mainly for emphasis, contrast, or clarity:

• Nos laeti sumus = We are happy (perhaps contrasting with someone else)

A very natural Latin sentence could also be:

• Laeti sumus, et multum laboramus.

Because laeti agrees with nos, which is plural.

• laetus = happy, singular masculine
• laeti = happy, plural masculine

Since the subject is we, the adjective must also be plural.

In Latin, adjectives agree with the noun or pronoun they describe in:

• number
• gender
• case

Here laeti matches nos.

Latin uses different adjective endings depending on gender.

• laeti = masculine plural
• laetae = feminine plural

So:

• Nos laeti sumus = We are happy if the group is male or mixed
• Nos laetae sumus = We are happy if the group is all female

A mixed group normally takes the masculine plural in Latin.

This is a very common Latin pattern:

• sumus = we are
• laeti = happy

Together, laeti sumus means we are happy.

Here laeti is a predicate adjective: it describes the subject after a form of esse (to be).

So this is like:

• Puella laeta est = The girl is happy
• Nos laeti sumus = We are happy

Sumus is the present indicative of esse (to be), first person plural.

So it means:

• we are

Some related forms are:

• sum = I am
• es = you are
• est = he/she/it is
• sumus = we are
• estis = you all are
• sunt = they are

Laboramus is present indicative active, first person plural, from laborare (to work).

It means:

• we work
• or sometimes we are working

Latin present tense can cover both of those English ideas, depending on context.

Because here multum is being used adverbially, meaning:

• much
• a lot
• greatly

It modifies the verb laboramus:

• multum laboramus = we work a lot

This is different from multus, multa, multum as an adjective meaning much/many when it describes a noun.

Compare:

• multum laboramus = we work a lot
• multum vinum = much wine
• multi homines = many people

So in your sentence, multum is not describing a noun; it is modifying the action.

Yes, Latin has more than one way to express a lot or very much with a verb.

• multum laboramus is perfectly normal and straightforward.
• valde often means strongly, greatly, or very, and is more common with adjectives or certain verbs.

For this sentence, multum laboramus is a very natural way to say we work a lot.

Because once the subject is clear, Latin often leaves it understood.

So:

• Nos laeti sumus, et multum laboramus

naturally means:

• We are happy, and we work a lot

Latin does not usually repeat the subject unless there is a reason to emphasize it:

• Nos laeti sumus, et nos multum laboramus would sound more emphatic or contrastive

Et means and.

It joins the two parts of the sentence:

• Nos laeti sumus = We are happy
• multum laboramus = we work a lot

So the full sentence is:

• We are happy, and we work a lot

Et is one of the most common Latin conjunctions.

No. Latin word order is fairly flexible because the endings show the grammar.

So these could all mean roughly the same thing:

• Nos laeti sumus, et multum laboramus.
• Laeti sumus, et multum laboramus.
• Nos sumus laeti, et laboramus multum.

The exact order can affect emphasis or style, but not the basic meaning.

In your sentence:

• nos is emphasized by being stated
• laeti sumus is a neat, natural phrase
• multum comes before laboramus, which is also very normal

Nos is nominative here, because it is the subject of the sentence.

• nos = we / us, depending on case and use

In this sentence it means we, so it is the subject form.

A learner may notice that nos can also appear as us in English translations in other contexts, but here the verb forms make it clear that nos is the subject.

Yes. It comes from laborare, which is a first-conjugation verb.

You can see that from the dictionary form ending in -are.

A few present tense forms are:

• laboro = I work
• laboras = you work
• laborat = he/she/it works
• laboramus = we work
• laboratis = you all work
• laborant = they work

So laboramus follows a very common first-conjugation pattern.

That is mostly a matter of punctuation style.

The sentence has two coordinated parts:

• Nos laeti sumus
• et multum laboramus

A comma can be used to make the structure clearer, especially in teaching materials. But Latin punctuation was not originally standardized the way modern punctuation is.

So the sentence could also appear without a comma:

• Nos laeti sumus et multum laboramus.

Both are fine in modern editions, depending on editorial style.