A mi prima también le da pereza madrugar los lunes, aunque luego nunca falta a clase.

Because Spanish very often uses indirect object doubling.

• le = to her
• a mi prima = clarifies exactly who her is

So A mi prima también le da pereza... literally looks like:

• To my cousin, it also gives laziness...

That sounds odd in English, but in Spanish it is normal and very common.

With verbs and expressions like gustar, interesar, dar pereza, dar igual, etc., the pronoun (me, te, le, nos...) is usually the core part, and the a + person phrase is added for clarity or emphasis.

So:

• Le da pereza madrugar = She finds getting up early a drag
• A mi prima le da pereza madrugar = My cousin finds getting up early a drag

Dar pereza is a very common expression meaning:

• to feel like a drag
• to make someone feel lazy about doing something
• to make someone not feel like doing something

So:

• Le da pereza madrugar does not mean she is a lazy person in general.
• It means getting up early feels like a chore to her or she can’t be bothered to get up early.

Compare:

• Es perezosa = she is lazy (a personal trait)
• Le da pereza madrugar = getting up early feels like a nuisance to her (a situation)

Because in Spanish, an infinitive can act like a noun.

Here, madrugar means something like:

• getting up early
• rising early

So the thing that gives pereza is the action itself:

• Madrugar los lunes le da pereza
• Getting up early on Mondays is a drag for her

This is very common in Spanish:

• Me gusta leer = I like reading / I like to read
• Nos cuesta estudiar = studying is hard for us
• Le da pereza salir = going out feels like a drag to her

They are similar, but madrugar is a bit more specific.

• madrugar = to get up very early, often earlier than you would like
• levantarse temprano = to get up early

So madrugar often suggests an unpleasantly early start.

Examples:

• Tengo que madrugar mañana = I have to get up very early tomorrow
• Me levanto temprano = I get up early

In your sentence, madrugar fits well because Mondays often involve an early, unwelcome start.

Because the subject is the whole action madrugar los lunes, and that whole idea is treated as singular.

So:

• Madrugar los lunes = one activity / one general thing
• therefore: da pereza

It works like English:

• Getting up early on Mondays is annoying not
• Getting up early on Mondays are annoying

Because Spanish usually uses the definite article with days of the week for habitual actions.

• los lunes = on Mondays / every Monday
• el lunes = on Monday (one specific Monday, depending on context)

So here:

• madrugar los lunes = getting up early on Mondays / every Monday

You do not normally say en los lunes for this meaning.

More examples:

• Trabajo los martes = I work on Tuesdays
• Vamos al gimnasio los viernes = We go to the gym on Fridays

También means also or too.

Here it suggests that my cousin too has this feeling, probably in addition to someone else already mentioned.

So the idea is:

• My cousin also finds getting up early on Mondays a drag

Its position is flexible in Spanish, but here A mi prima también... sounds natural and clearly connects the also to my cousin / her situation.

Aunque means although / even though and introduces a contrast.

So:

• ...aunque luego nunca falta a clase = ...although she later never misses class

This sets up a nice contrast:

• she hates getting up early,
• but she still always goes to class.

Could pero work? Sometimes yes, but it changes the structure a bit:

• A mi prima también le da pereza madrugar los lunes, pero luego nunca falta a clase.

That is also grammatical and natural.
The difference is mainly one of style and nuance:

• aunque = more like although/even though
• pero = more like but

In this sentence, aunque sounds slightly more elegant and tightly linked.

Here luego means something like:

• then
• afterwards
• later
• sometimes even all the same, in the end depending on context

So the sentence means:

• getting up early feels like a drag,
• but then / afterwards / all the same she never misses class

In Spain, luego is very common in this kind of contrast.

Important: it does not always mean a long time later. Often it just marks the next step in the situation.

Because the verb faltar, when it means to miss or to be absent from, often uses a before the place/event/activity.

So:

• faltar a clase = to miss class / be absent from class
• faltar al trabajo = to miss work
• faltar a una reunión = to miss a meeting

In your sentence:

• nunca falta a clase = she never misses class / she is never absent from class

This is just the normal pattern of the verb here.

Because faltar a clase is a very common idiomatic expression meaning to miss class / skip class / be absent from lessons in a general sense.

Even if someone has several lessons, Spanish often uses singular clase when talking about school attendance generally.

So:

• Nunca falta a clase = She never misses class / she never skips class

You may also hear a clases in some contexts or regions, but a clase is extremely common and natural.

Yes. The present tense here expresses something habitual or generally true.

• le da pereza = it is a regular feeling she has
• nunca falta = she habitually never misses class

This is why los lunes fits well too: it signals repetition.

So the sentence is not about just one Monday; it describes a usual pattern.

Yes, to some extent. Spanish word order is flexible, though some versions sound more natural than others.

For example:

• A mi prima también le da pereza madrugar los lunes...
• A mi prima le da también pereza madrugar los lunes...
• También a mi prima le da pereza madrugar los lunes...

These are all possible, but the original sounds very natural.

The original order works well because it introduces:

1. who we are talking about: A mi prima
2. the idea of also: también
3. the pronoun required by the structure: le
4. the thing that feels like a drag: madrugar los lunes

So even if alternatives are possible, the given version is a very good standard one.