Learn Hardcore Portuguese (Portugal): Eu vejo o caminho da janela do quarto.

Vejo o caminho da janela do quarto. – Completely correct and very natural.
Eu vejo o caminho da janela do quarto. – Also correct, but adds emphasis to I.

You typically include eu when:

you contrast with someone else:
Eu vejo o caminho, mas tu não vês. – I see the path, but you don’t.
you really want to stress the subject:
Eu é que vejo o caminho. – I am the one who sees the path.

In a neutral sentence, most European Portuguese speakers would simply say:
Vejo o caminho da janela do quarto.

Why is it o caminho and not just caminho?

Portuguese uses definite articles (o / a / os / as) much more than English.

o caminho = the path
caminho without an article is much less common in this kind of sentence and would sound incomplete or ungrammatical here.

You normally drop the article when:

you speak very generally, like a concept:
Caminho é vida. – Path is life (very literary / poetic).
in some fixed expressions (e.g., ir a pé, ter razão).

In your sentence, you are talking about a specific, visible path, so o caminho is the normal, natural choice.

What exactly does caminho mean here? Is it like street, road, or way?

Caminho is a general word meaning path / way. It can be:

a literal path (footpath, track):
Vejo o caminho de terra. – I see the dirt path.
a route or way:
Conheço o caminho para a escola. – I know the way to school.
a figurative way/direction:
Este é o melhor caminho para o sucesso. – This is the best path to success.

Contrast with:

rua – street (inside a town/city)
estrada – road (often for cars, between places)
caminhada – walk, hike (the activity)
jeito / maneira / forma – way, method (in a more abstract sense)

In Eu vejo o caminho da janela do quarto, we naturally understand a physical path or route that can be seen from the window.

What does da mean in da janela? Why not just de janela?

Da is a contraction of de + a:

de = of / from
a = the (feminine singular)
janela = window (feminine)

So:

de + a janela → da janela = from the window / of the window

In standard Portuguese, when de is followed by a definite article, you must contract:

de + o quarto → do quarto
de + a janela → da janela
de + os quartos → dos quartos
de + as janelas → das janelas

Saying de janela without the article would sound wrong here in European Portuguese. You need da janela.

Does da janela do quarto mean from the bedroom window or of the bedroom’s window? How should I understand this structure?

Here, da janela do quarto is a single phrase meaning from the bedroom window.

Breakdown:

da janela – from the window / of the window
do quarto – of the bedroom (literally of the room)

So:

da janela do quarto = from the window of the bedroom = from the bedroom window

Even though de often means of, in this kind of context da janela is naturally read as from the window (a point of view or starting point), not as possession.

We understand:

Eu vejo o caminho [a partir] da janela do quarto.
I see the path from the bedroom window.

It would be very unusual to interpret it as the path of the bedroom’s window; that makes no sense in Portuguese, so context forces the correct reading.

Why is it do quarto and not do meu quarto if I mean from my bedroom window?

In Portuguese, possession is often understood from context and does not need to be marked with meu / minha every time.

So:

da janela do quarto often naturally implies from the bedroom window (of the place we’re talking about, probably mine / ours).

Compare:

Estou no quarto. – I’m in the bedroom. (Usually understood as my bedroom if we’re talking about my house.)
A janela do quarto é grande. – The bedroom window is big.
Vejo o jardim da janela do quarto. – I see the garden from the bedroom window.

If you want to be explicit:

da janela do meu quarto – from my bedroom window
da janela do teu / seu / nosso quarto – from your / his/her / our bedroom

All are correct; adding meu just makes the owner explicit. Without it, context usually fills in that detail.

Could the sentence also be Eu vejo o caminho pela janela do quarto? What is the difference between da janela and pela janela?

Both are possible, but there is a subtle difference:

da janela do quarto
Focus: the point of view / origin – from the bedroom window
→ I am at the window; from there I see the path.
pela janela do quarto
pela = por + a (through / by the)
Focus: the medium / opening – through the bedroom window
→ I use the window as an opening to see the path.

In many everyday contexts they can overlap, and both can be translated as from the bedroom window, but:

da janela sounds more like from that vantage point.
pela janela highlights that you look through the window as an opening.

Could I move da janela do quarto earlier and say Eu vejo da janela do quarto o caminho?

Yes, that word order is grammatically correct:

Eu vejo o caminho da janela do quarto. – neutral, most common.
Eu vejo da janela do quarto o caminho. – still correct, a bit more marked in terms of rhythm and emphasis.

Portuguese word order is relatively flexible. Moving da janela do quarto forward can:

sound slightly more emphatic or stylistic
put more focus on from the bedroom window as the important information

In everyday speech, the original order (vejo o caminho da janela do quarto) is more natural.

Why is it quarto? Does quarto mean room or bedroom?

In European Portuguese:

quarto very commonly means bedroom.
It can also mean room in general, depending on context, but in everyday speech quarto = bedroom is the default.

Examples:

Vou para o quarto. – I’m going to the bedroom.
O quarto das crianças é pequeno. – The children’s bedroom is small.

Quarto can also mean fourth (the ordinal number), but:

when it means fourth, it is an adjective: o quarto andar – the fourth floor.
in your sentence, as a noun after do, it is read as (bed)room.

How is vejo formed? What is the infinitive of this verb, and what tense is it?

Vejo is the 1st person singular – present indicative of the verb ver (to see).

Conjugation in the present (European Portuguese pronunciation in brackets):

eu vejo – I see (VE-joo)
tu vês – you see (singular, informal) (VESH)
ele / ela vê – he / she sees (VEH)
nós vemos – we see (VEH-moosh / VEH-mush)
vocês veem – you (plural) see (VE-êm)
eles / elas veem – they see (VE-êm)

So Eu vejo o caminho da janela do quarto is in the present tense, describing something that happens now or habitually.

What is the difference between ver and olhar? Could I say Eu olho o caminho da janela do quarto?

Ver and olhar are related but not interchangeable:

ver = to see (the perception, often passive)
- Vejo o caminho. – I see the path.
olhar = to look (at) (an intentional action)
- Olho para o caminho. – I look at the path.

Important points:

olhar normally needs para when you say what you look at:
- Olho para a rua. – I look at the street.
- Olho pela janela. – I look through the window.
Eu olho o caminho (without para) sounds odd or wrong in European Portuguese.

For your original idea:

Eu vejo o caminho da janela do quarto. – natural and correct.
Eu olho para o caminho da janela do quarto. – grammatically possible but less natural; it sounds like you’re deliberately looking at the path from the bedroom window.

Most of the time, to describe what is visible, you use ver, not olhar.

Why do we have o caminho, a janela, and o quarto? How do I know which article and gender to use?

Each noun in Portuguese has a fixed grammatical gender:

caminho – masculine → o caminho
janela – feminine → a janela
quarto – masculine → o quarto

General tendencies (not strict rules):

nouns ending in -o are often masculine: o caminho, o quarto, o carro
nouns ending in -a are often feminine: a janela, a casa, a porta

But there are many exceptions, so you must learn each noun with its article:

o problema (masculine, even though it ends in -a)
a foto (feminine, short for a fotografia)

In your sentence, the articles must agree with the noun’s gender:

o caminho da janela do quarto
- o → masculine singular (caminho)
- da → de + a (janela, feminine)
- do → de + o (quarto, masculine)

Could this sentence ever mean I see the way from the window to the bedroom?

No, not as it stands. The structure da janela do quarto clearly groups janela and quarto together:

da janela do quarto = from the bedroom window

To talk about the path from the window to the bedroom, you would need to express both ends explicitly, for example:

Vejo o caminho da janela até ao quarto.
I see the path from the window to the bedroom.

Here:

da janela – from the window
até ao quarto – to the bedroom

In your original sentence, do quarto is attached to janela, not to caminho or another preposition of movement. So it is understood as bedroom window, not a path from X to Y.