Pluvia nos domi manere cogit, sed pater dicit otium quoque utile esse.

Here nos is accusative, not nominative.

That is because nos is the object of cogit: the rain forces us.
Latin often uses this pattern with cogere:

• cogere + accusative + infinitive
• aliquem aliquid facere cogere = to force someone to do something

So:

• Pluvia = the rain
• nos = us
• manere = to stay
• cogit = forces

Literally: The rain forces us to stay...

At the same time, nos is also understood as the subject of manere. In English we say forces us to stay; Latin does the same idea with accusative + infinitive.

Because after cogit Latin uses an infinitive to express the action someone is forced to do.

So:

• nos manere cogit = forces us to stay

This is a very common construction with verbs like iubet, vetat, cogit, and similar verbs. English often uses to + verb, and Latin uses the infinitive:

• to stay = manere

Domi is a special form meaning at home. It is called the locative.

A few place words in Latin have special forms for location, and domus is one of them:

• domi = at home
• domum = to home, homeward
• domo = from home

So:

• domi manere = to stay at home

You can sometimes see in domo, but domi is the normal compact way to say at home.

After verbs of saying, thinking, knowing, hearing, and so on, Latin very often uses indirect statement.

The normal pattern is:

• accusative subject + infinitive verb

So in this sentence:

• pater dicit = father says
• otium ... utile esse = that leisure is useful

This is one of the most important Latin constructions to learn.
English often uses that; Latin usually uses accusative + infinitive instead.

Because in an indirect statement, the subject goes into the accusative.

So in:

• pater dicit otium utile esse

the word otium is the logical subject of esse. It is the thing that is said to be useful. But because it is inside indirect statement after dicit, Latin puts it in the accusative.

One detail can be confusing: otium is neuter singular, and neuter nouns often have the same form in nominative and accusative. So even though it looks like a nominative, its role here is accusative.

Because utile agrees with otium, which is neuter singular.

Utilis, utile is a third-declension adjective:

• masculine/feminine singular: utilis
• neuter singular: utile

Since otium is neuter, the predicate adjective must also be neuter:

• otium utile esse = leisure to be useful / that leisure is useful

Even inside indirect statement, the adjective still agrees with the noun it describes.

Quoque means also or too.

It is often placed right after the word it emphasizes. So:

• otium quoque = leisure too / leisure also

That suggests something like not only work or activity, but leisure too.

This placement is very normal in Latin. Compare:

• ego quoque = I too
• Romani quoque = the Romans too

So the position of quoque helps show what is being marked as also.

Because Latin word order is flexible, and infinitives like esse often come at the end of their clause.

So:

• otium quoque utile esse

puts the key idea useful to be at the end of the indirect statement.

Latin does not rely on word order as strictly as English does, because the endings show grammatical relationships. The writer can move words around for emphasis, rhythm, or style.

Sometimes Latin can omit a form of esse, especially in poetry or very compressed style. But in prose, and especially in an indirect statement, it is very normal to include it.

Here utile esse is the full standard expression:

• to be useful
• that it is useful

So yes, omission can happen in some contexts, but in a sentence like this, esse is exactly what you would expect.

Latin word order is much freer than English word order because Latin uses case endings and verb forms to show who is doing what.

A very literal arrangement would be:

• Rain us at home to stay forces, but father says leisure also useful to be

That sounds strange in English, but it is perfectly normal in Latin.

A few useful points here:

• Pluvia comes first for emphasis on the cause.
• domi is placed near manere, since they belong together.
• pater dicit introduces the second clause.
• esse comes at the end, which is common in indirect statement.

So when reading Latin, it helps not to expect English order. Instead, look for:

• the main verb
• noun endings
• infinitives
• clause markers like sed

Because Latin has no articles.

So a noun like pater can mean:

• father
• the father
• sometimes even a father

The exact English translation depends on context.

Likewise:

• pluvia = rain / the rain
• otium = leisure / rest / free time

When translating into English, you add the or a where it sounds natural, but Latin itself does not use separate words for them.