A mi primo le cuesta madrugar porque es bastante perezoso por la mañana.

Yes — they both point to the same person, and that is very normal in Spanish.

This is called clitic doubling. In sentences like this, Spanish often uses:

• the full phrase: a mi primo
• plus the indirect object pronoun: le

So:

• A mi primo le cuesta madrugar
  literally works like
• To my cousin, getting up early is difficult

The le is not optional in standard Spanish here. It helps mark the person affected by costar.

You will see this pattern a lot:

• A Juan le gusta el café
• A mi hermana le molesta el ruido
• A nosotros nos cuesta concentrarnos

Here costar means to be difficult for someone or to take effort.

The pattern is:

• A alguien le cuesta + infinitive

So:

• A mi primo le cuesta madrugar = It is hard for my cousin to get up early

Grammatically, the thing that is difficult is the infinitive action:

• madrugar

Other examples:

• Me cuesta entender esto = This is hard for me to understand
• Nos cuesta levantarnos temprano = It’s hard for us to get up early
• A Laura le cuesta hablar en público = It’s hard for Laura to speak in public

Because the subject of costar here is the whole action madrugar, and that is treated as singular.

So:

• Madrugar cuesta ...
• therefore: le cuesta madrugar

Compare:

• Le cuesta madrugar = Getting up early is hard for him/her
• Le cuestan las mañanas = Mornings are hard for him/her

In the second example, the subject is las mañanas, which is plural, so the verb becomes cuestan.

Both are possible, but they are not exactly identical.

• madrugar = to get up very early / to rise early
• levantarse temprano = to get up early

Madrugar is a very common Spanish verb and often implies earlier than one would like or very early in the day. It is a compact, natural choice here.

So:

• Le cuesta madrugar sounds very natural.
• Le cuesta levantarse temprano is also correct, but slightly more explicit.

It usually means to get up early or to rise early, not just to wake up.

That distinction matters:

• despertarse = to wake up
• levantarse = to get up
• madrugar = to get up early / start the day very early

So if someone opens their eyes at 6:00 but stays in bed, madrugar may not be the best verb. It usually suggests actually starting the day early.

Because porque is the conjunction meaning because.

Spanish distinguishes these forms:

• porque = because
• por qué = why
• porqué = the reason (a noun)
• por que = less common combination in specific structures

In your sentence:

• ... porque es bastante perezoso ...
  = ... because he is quite lazy ...

If you were asking a question, you would use por qué:

• ¿Por qué le cuesta madrugar? = Why is it hard for him to get up early?

Because ser is being used to describe a more general characteristic.

• es perezoso = he is lazy / he’s a lazy person
• está perezoso = he feels lazy / he is being lazy right now

In this sentence, the idea is that his difficulty getting up early is due to a personal trait, at least in the mornings. So es fits better.

That said, Spanish can sometimes use estar perezoso for a temporary state:

• Hoy está perezoso = He’s feeling lazy today

But here es bastante perezoso por la mañana sounds like a habitual description.

Here bastante means quite, rather, or fairly.

So:

• bastante perezoso = quite lazy

It softens the statement a little compared with something stronger like:

• muy perezoso = very lazy

Other uses of bastante:

• bastante bien = quite well
• bastante difícil = fairly difficult
• Tengo bastante tiempo = I have quite a lot of time / enough time

Depending on context, bastante can mean either quite/fairly or enough/a lot.

Because por + time expression is a very common way in Spanish to mean during that part of the day.

So:

• por la mañana = in the morning
• por la tarde = in the afternoon / evening
• por la noche = at night / in the evening

In Spain, por la mañana is the standard natural expression here.

English speakers sometimes expect en la mañana, but in Spain that is much less natural in this kind of sentence.

They are not the same.

• por la mañana = in the morning
• de mañana = in the morning / early, but in many varieties it is less common or more regional in this meaning

In Spain, por la mañana is the usual everyday choice.

Examples:

• Trabajo por la mañana = I work in the morning
• Estoy cansado por la mañana = I’m tired in the morning

So in your sentence, por la mañana is the most natural option.

Yes. Spanish word order is flexible.

The original sentence:

• A mi primo le cuesta madrugar porque es bastante perezoso por la mañana.

You could also say:

• Le cuesta madrugar a mi primo porque es bastante perezoso por la mañana.
• Porque es bastante perezoso por la mañana, a mi primo le cuesta madrugar.

But the original order is very natural because it introduces the person first.

Putting a mi primo at the beginning gives it a slight topic emphasis, like as for my cousin...

Because after costar, Spanish normally uses a bare infinitive.

Pattern:

• costar + infinitive

So:

• le cuesta madrugar
• me cuesta estudiar
• nos cuesta decidirnos

The infinitive in Spanish often functions like an English -ing form or verbal noun:

• madrugar = getting up early
• estudiar = studying
• decidirse = making up one’s mind

You would not normally say le cuesta el madrugar in neutral everyday Spanish.

Yes.

• primo = male cousin
• prima = female cousin

So if you were talking about a female cousin, you would say:

• A mi prima le cuesta madrugar porque es bastante perezosa por la mañana.

Notice that both the noun and adjective change:

• primo → prima
• perezoso → perezosa

Yes. Le is the indirect object pronoun for both him and her (and also you formal singular).

By itself, le cuesta could mean:

• it is hard for him
• it is hard for her
• it is hard for you (formal)

That is one reason Spanish often includes the full phrase:

• A mi primo le cuesta...
• A mi prima le cuesta...
• A usted le cuesta...

The added phrase removes ambiguity.

Perezoso is a standard, neutral word for lazy and works very well here.

Other possibilities exist, but they can sound different:

• vago = lazy, but can sound more colloquial and sometimes harsher
• flojo = weak / lazy / lacking energy, depending on context

So:

• perezoso is a safe, natural choice
• vago may sound more blunt
• flojo changes the nuance

For a learner, perezoso is the best straightforward match here.