Kada pada kiša, ostajem kod kuće i čitam knjigu.

Yes. Padati literally means to fall, so kiša pada is literally rain is falling. That is the normal Serbian way to say it is raining.

Unlike English, Serbian does not need a dummy subject like it. The real subject is kiša (rain), so the sentence is built around that:

• kiša pada = rain falls / it rains

This is a very common weather expression.

Yes. Kada and kad both mean when.

• kada is a bit more full/formal/neutral
• kad is very common in everyday speech and writing

So these are both natural:

• Kada pada kiša, ostajem kod kuće.
• Kad pada kiša, ostajem kod kuće.

The meaning does not really change here.

In Serbian, the present tense is often used for habitual or general repeated actions, just like in English:

• When it rains, I stay home and read a book.

So:

• pada = it rains / is raining
• ostajem = I stay
• čitam = I read

This sentence does not usually mean one specific single event. It sounds like a general habit or usual pattern.

This is about aspect, which is very important in Serbian.

• ostajati / ostati = to stay / remain
• ostajem comes from the imperfective verb ostajati
• ostanem comes from the perfective verb ostati

In a habitual sentence like this, Serbian normally uses the imperfective:

• Kada pada kiša, ostajem kod kuće.
  = When it rains, I stay at home.

Using ostanem would not sound right here for a general repeated habit. Perfective verbs usually focus on a completed single event or result, not a regular repeated situation of this kind.

Because Serbian verb forms already show the person clearly.

• ostajem = I stay
• čitam = I read

So the pronoun ja (I) is usually unnecessary unless you want emphasis or contrast.

For example:

• Kada pada kiša, ostajem kod kuće. = normal
• Kada pada kiša, ja ostajem kod kuće. = emphasizes I (maybe unlike someone else)

English usually needs the subject pronoun, but Serbian often drops it.

Kod kuće is the normal Serbian expression for at home.

• kod kuće = at home
• u kući = in the house

These are close, but not always identical.

Kod kuće talks about being at one’s home in a general sense.
U kući focuses more on physical location inside the building.

So in this sentence, kod kuće is the most natural choice:

• Kada pada kiša, ostajem kod kuće. = When it rains, I stay at home.

In the expression kod kuće, the noun kuća changes form because kod requires the genitive case.

• dictionary form: kuća = house / home
• genitive singular: kuće

So:

• kod + genitive
• kod kuće

This is a fixed and very common expression, so it is worth learning as a whole phrase.

Because knjigu is the accusative singular, and it is the direct object of čitam.

• knjiga = nominative singular (a book / the book as subject)
• knjigu = accusative singular (a book / the book as object)

Since you are reading the book, the book is the object:

• Čitam knjigu. = I am reading a book / the book.

This is a very common pattern with feminine nouns ending in -a:

• nominative: knjiga
• accusative: knjigu

Both are possible.

• čitam = I am reading / I read
• čitam knjigu = I am reading a book / the book

Adding knjigu makes the sentence a bit more complete and concrete. Without it, the meaning is still clear, but more general.

So:

• Kada pada kiša, ostajem kod kuće i čitam. = natural
• Kada pada kiša, ostajem kod kuće i čitam knjigu. = also natural, slightly more specific

It can mean either, depending on context.

Serbian does not have articles like English a and the, so knjigu by itself could mean:

• a book
• the book

In this sentence, English would probably translate it as a book unless the context has already identified a specific book.

So Serbian leaves that distinction to context.

Because Kada pada kiša is a subordinate clause placed at the beginning of the sentence.

Structure:

• Kada pada kiša = When it rains
• ostajem kod kuće i čitam knjigu = main clause

When this kind of when-clause comes first, Serbian normally separates it with a comma:

• Kada pada kiša, ostajem kod kuće...

This is similar to English:

• When it rains, I stay at home...

Yes, Serbian word order is fairly flexible, though some versions sound more neutral than others.

The given sentence is very natural and neutral:

• Kada pada kiša, ostajem kod kuće i čitam knjigu.

You could also say:

• Ostajem kod kuće i čitam knjigu kada pada kiša.

That version is also grammatical, but it places the when part later and may sound slightly different in emphasis.

So the original sentence is a very good standard version for learners.

It has two main parts:

1. Kada pada kiša

   • kada = when
   • kiša = rain
   • pada = falls / rains

2. ostajem kod kuće i čitam knjigu

   • ostajem = I stay
   • kod kuće = at home
   • i = and
   • čitam = I read
   • knjigu = a/the book

So the pattern is:

When + event, I do X and do Y.

This is a very common and useful sentence pattern in Serbian.