Des de fa anys, la meva mare llegeix el diari a internet cada matí.

It is a very common Catalan way to express a duration that started in the past and continues up to now.

• des de fa + time expression = for + time expression
• des de fa anys = for years

So in this sentence, it tells you that this has been her habit for a long time and is still true now.

Because Catalan often uses the present tense for actions or habits that began in the past and still continue now, especially with expressions like des de fa anys.

English often prefers:

• has read
• has been reading

Catalan can simply use the present:

• Des de fa anys, la meva mare llegeix...

So this is normal Catalan, not a mistake.

In standard Catalan, possessives usually go together with the definite article:

• el meu pare
• la meva mare
• el nostre cotxe

So la meva mare is the normal standard form. English does not use an article here, but Catalan usually does.

The article may be left out in some special cases, such as:

• direct address: Mare, vine!
• some dialectal or literary forms

But in an ordinary sentence like this one, la meva mare is exactly what you would expect.

Llegeix is the third person singular present indicative of llegir (to read).

It matches la meva mare:

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

So here llegeix means she reads.

Because Catalan normally uses the definite article in this kind of expression.

• llegir el diari = to read the newspaper / the paper

This is idiomatic Catalan. Even when English might sound more general, Catalan still often uses el.

So el diari here does not necessarily mean one specific newspaper you have already identified in the conversation. It is simply the normal way to say the activity.

In current Catalan, a internet is a normal way to say on the internet.

Catalan can treat internet without an article here, so you get:

• a internet = on the internet

You may also hear:

• per internet = via the internet / over the internet

The difference is small in many everyday contexts, but a internet fits very naturally in this sentence.

Yes. Both are natural.

• cada matí = every morning
• tots els matins = every morning / all mornings

The sentence could also be:

• Des de fa anys, la meva mare llegeix el diari a internet tots els matins.

That said, cada matí is slightly more compact and perfectly natural here.

Catalan often leaves out subject pronouns because the verb form already gives the person and number, or because the subject is already stated.

Here the subject is explicitly present:

• la meva mare

So there is no need to add ella.

If you did add ella, it would usually give extra emphasis or contrast:

• La meva mare, ella llegeix el diari...

That is possible, but not the neutral version.

No. Catalan word order is fairly flexible.

The version you have:

• Des de fa anys, la meva mare llegeix el diari a internet cada matí.

puts the long-time duration first, which gives it emphasis: for years now...

You could also say:

• La meva mare llegeix el diari a internet cada matí des de fa anys.

That is also grammatical. The meaning stays basically the same, but the focus changes a little.

Because cada is normally followed by a singular noun in Catalan:

• cada dia
• cada setmana
• cada matí

So cada matí is the correct pattern, just like English every morning uses a singular noun.