El meu pare llegeix el diari a internet quan arriba a casa.

In standard Catalan, possessives like meu, meva, meus, meves usually go together with the definite article:

• el meu pare = my father
• la meva mare = my mother
• els meus llibres = my books

So el meu pare is the normal standard pattern. English does not use the here, but Catalan usually does.

Llegeix is the 3rd person singular present of llegir:

• jo llegeixo = I read
• tu llegeixes = you read
• ell/ella llegeix = he/she reads

So here it means he reads.

Because llegir is an incoative -ir verb. Many Catalan -ir verbs add -eix- in several present-tense forms:

• llegeixo
• llegeixes
• llegeix
• llegeixen

But not all -ir verbs do this, and even this verb does not keep -eix- in every form:

• llegim
• llegiu

So this is a normal conjugation pattern, not an extra word part with a separate meaning.

El diari means the newspaper. The definite article makes it specific.

In a sentence like this, Catalan often uses el diari for a regular, familiar activity: reading the newspaper/news. If you said un diari, it would sound more like a newspaper or one newspaper, which is less specific.

Here a internet means on the internet or online.

Catalan often uses a internet for this idea. You may also hear per internet, especially when the meaning is more like via the internet. In this sentence, a internet is a natural way to say that the reading happens online.

Catalan often leaves out subject pronouns because the verb ending already tells you the person.

• arriba already means he/she arrives

So quan arriba a casa naturally means when he gets home. Adding ell is possible, but it usually adds emphasis or contrast.

A casa is the normal idiomatic way to say home.

• arriba a casa = he gets home

If you say a la casa, you are talking more literally about the house/building, not the idea of home. This is similar to the difference between go home and go to the house in English.

Because the sentence describes a habit or routine.

• llegeix = reads
• arriba = arrives / gets

So the sentence means something like My father reads the newspaper online when he gets home. Catalan, like English, often uses the present tense for repeated actions.

Yes. In this sentence, the understood subject of arriba is the same as the subject of the main clause: el meu pare.

So the structure is:

• El meu pare ... quan arriba a casa
• My father ... when he gets home

Catalan often omits repeated subjects when they are clear from context.

Because when the time clause comes after the main clause, Catalan normally does not use a comma:

• El meu pare llegeix el diari a internet quan arriba a casa.

If you put the time clause first, a comma is much more natural:

• Quan arriba a casa, el meu pare llegeix el diari a internet.

Both word orders are correct.