Learn Hardcore Portuguese (Portugal): Lavo os lençóis ao sábado para o quarto cheirar sempre a limpo. - I wash the sheets on Saturdays so that the bedroom always smells clean.

Lavo ends in -o, which clearly marks 1st person singular (I).
So Lavo os lençóis already means I wash the sheets; eu is usually unnecessary.
You add eu mainly for emphasis or contrast:
- Eu lavo os lençóis ao sábado. = I wash the sheets on Saturdays (not someone else).

What exactly does ao sábado mean, and how is it different from no sábado or aos sábados?

Ao sábado means on Saturdays, as a regular habit (every or most Saturdays).

ao sábado
- Literally: on the Saturday, but in European Portuguese it normally means on Saturdays (habitual).
- Very common in Portugal for routines:
  - Vou ao ginásio ao sábado. = I go to the gym on Saturdays.
aos sábados
- Also means on Saturdays (regularly).
- Slightly more explicit that it’s plural, but in practice it’s very close in meaning to ao sábado.
no sábado
- Means on Saturday referring to one specific Saturday (often context/this coming one, last one, etc.):
  - No sábado vou lavar os lençóis. = On Saturday I’m going to wash the sheets (this particular Saturday).

So in your sentence, ao sábado tells us it’s a habit, not a one‑off event.

Is lençóis the normal word for “bedsheets” in Portugal, and what is its singular and gender?

Yes, lençóis is the standard word for bed sheets in Portugal.

Singular: o lençol (masculine) = the sheet
Plural: os lençóis (masculine) = the sheets
The stress moves: len‑ÇOL → len‑ÇÓIS.
A common phrase is os lençóis da cama = the bed sheets.
A more general term is a roupa de cama = bedding / bed linen (includes sheets, pillowcases, etc.).

Why do we use para in para o quarto cheirar… and not por?

Here para introduces a purpose: in order for the room to smell…

para is used for:
- purpose / goal:
  - Estudo para aprender. = I study to learn.
  - Lavo os lençóis ao sábado para o quarto cheirar sempre a limpo.
    I wash the sheets on Saturdays so that the room always smells clean.
por is more about cause, reason, means, duration, movement through:
- Fiz isto por ti. = I did this because of you / for you.
- Passeei por Lisboa. = I walked around Lisbon.

So por o quarto cheirar would be wrong here; you want the purpose, so you use para.

Why is it para o quarto cheirar (infinitive) and not something like para o quarto cheire?

Portuguese has two common ways to express “so that the room smells…”:

para + infinitive with an explicit subject (very natural in everyday European Portuguese):
- Lavo os lençóis ao sábado para o quarto cheirar sempre a limpo.
  Literally: I wash the sheets on Saturdays for the room to always smell clean.
para que + subjunctive (slightly more formal or textbook-like):
- …para que o quarto cheire sempre a limpo.

What’s happening:

Lavo is the main verb (finite, present tense).
cheirar is an infinitive verb in a subordinate purpose clause.
o quarto is the subject of the infinitive (the room is doing the smelling).

So both are grammatically acceptable:

para o quarto cheirar sempre a limpo (very normal in speech, especially in Portugal)
para que o quarto cheire sempre a limpo (more “school‑book” style)

In your sentence, the choice of the infinitive sounds natural and colloquial, not incorrect.

What does cheirar mean here exactly? Is it “to smell (sniff)” or “to smell like / to smell of”?

Here cheirar means “to smell of / to smell like”, i.e. to give off a smell.

Portuguese distinguishes two main uses:

cheirar a + noun / adjective = to smell like / of (emit a smell)
- O quarto cheira a limpo. = The room smells clean / of cleanliness.
- Cheira a café. = It smells of coffee.
- Cheira a fumo. = It smells of smoke.
cheirar + object = to smell something (sniff it)
- Cheirei as flores. = I smelled the flowers.
- Cheira isto. = Smell this.

In your sentence, it’s clearly the first meaning:

o quarto cheirar sempre a limpo = for the room always to smell clean / fresh.

How does sempre work in this sentence, and what does it mean here?

In this sentence sempre means “always”:

…para o quarto cheirar sempre a limpo.
= …so that the room always smells clean.

A few points:

Normal position: after the verb it modifies.
- cheirar sempre = always smell
You can move it for emphasis:
- …para o quarto sempre cheirar a limpo. – possible, but more emphatic/unusual.
sempre can also mean things like “still / after all” in other contexts, but not here:
- Ele veio sempre. = He ended up coming after all.

In this sentence, only the “always” reading makes sense.

What does the expression a limpo mean, and why is there an a before limpo?

In cheirar a limpo, the a is a preposition that goes with cheirar:

cheirar a X = to smell of / like X.

Here, limpo is the adjective clean, but used almost like a noun (cleanliness / something clean). So:

cheirar a limpo ≈ to smell of cleanliness / to smell clean / to smell fresh.

Some parallels:

cheirar a novo = to smell new
cheirar a gasolina = to smell of petrol
cheirar a pão fresco = to smell of fresh bread

So:

a is not an article; it’s the preposition required by cheirar a.
limpo stays masculine singular because it’s used in a general, abstract sense (like cleanliness), not agreeing with quarto directly.

Together, a limpo is best learned as part of the fixed expression cheirar a limpo.

Could I say cheirar limpo instead of cheirar a limpo?

For the literal smell of a room or laundry, native speakers normally say cheirar a limpo.

cheirar a limpo = to smell clean / smell of cleanliness (your sentence).
cheirar limpo exists, but it’s usually figurative:
- Esse negócio não cheira limpo. = There’s something fishy about that deal / It doesn’t look honest.

So for your meaning (smell freshly washed, smell nice and clean), you should keep the a:

…para o quarto cheirar sempre a limpo. ✔
…para o quarto cheirar sempre limpo. ✖ (would sound odd or be read figuratively).

Is the word order in para o quarto cheirar sempre a limpo fixed, or can I move sempre around?

The most natural word order is exactly what you have:

para o quarto cheirar sempre a limpo

Other possibilities:

para o quarto sempre cheirar a limpo – possible, but feels more marked/emphatic and less neutral.
para o quarto cheirar a limpo sempre – grammatically possible, but less usual.

General rules:

Keep cheirar and a limpo together (or almost together).
Place sempre close to the verb it modifies (cheirar), normally right after it.

So for everyday speech and writing, para o quarto cheirar sempre a limpo is the best choice.

Is this sentence particularly European Portuguese? How might a Brazilian Portuguese speaker say something similar?

The sentence is very natural European Portuguese, mainly because of ao sábado.

In Portugal:
- Lavo os lençóis ao sábado para o quarto cheirar sempre a limpo.

In Brazil, people would almost never say ao sábado. More likely:

Lavo os lençóis todo sábado pra o quarto cheirar sempre a limpo.
Or with slightly different phrasing:
- Lavo os lençóis todo sábado pra o quarto ficar sempre cheirando a limpeza / cheirosinho.

Notes:

todo sábado (Br) ≈ ao sábado / aos sábados (Pt).
cheirar a limpo exists in Brazil too, but you also hear:
- cheirar a limpeza, cheiro de limpeza, estar cheirosinho, etc.

So the structure is understood in both varieties, but the choice of day expression is strongly European.

How could I make this sentence a bit more formal or explicit?

A slightly more formal or “textbook” version would use para que + subjunctive and maybe a fuller time phrase:

Costumo lavar os lençóis ao sábado, para que o quarto cheire sempre a limpo.
= I usually wash the sheets on Saturdays so that the room always smells clean.

Even more formal:

Lavo os lençóis todos os sábados, de modo a que o quarto cheire sempre a limpo.
= I wash the sheets every Saturday, so that the room will always smell clean.

Your original sentence is perfectly correct and natural; these versions just sound a bit more formal / written.