Learn Hardcore Portuguese (Portugal): Hoje à noite quero fazer um passeio longo depois do jantar. - Tonight I want to go for a long walk after dinner.

Putting them together, Hoje à noite literally means today at night, which corresponds to tonight.
Na noite de hoje is grammatically possible but sounds very formal or unnatural in normal conversation.

What exactly is happening in à noite? Why is there a grave accent (crase) on à?

À in à noite is a contraction of:

the preposition a (to, at)
plus the definite article a (the) before a feminine noun (noite)

So: a (prep.) + a (article) + noite → à noite

The grave accent shows this contraction (called crase).
Literally, à noite means at the night / in the evening, but idiomatically it just means at night / in the evening.

Could I say na noite instead of à noite?

You can say na noite, but it usually has a different feel:

à noite = in the evening / at night (general time of day)
na noite = in the night, often with a more specific or descriptive sense, e.g.
- Na noite de Natal – on Christmas Eve / on the night of Christmas
- Na noite de sábado – on Saturday night

In your sentence, the natural choice is Hoje à noite, not Hoje na noite.

Can I also say Esta noite instead of Hoje à noite? Is there a difference?

Yes, both are possible:

Hoje à noite – very clearly tonight (this night of today)
Esta noite – this night / tonight, but slightly looser in relation to “today”

In most everyday contexts, they overlap and both are fine. In European Portuguese:

Hoje à noite is very common and explicit.
Esta noite is also common, sometimes a bit more neutral or context‑dependent.

In your sentence, both Hoje à noite and Esta noite would sound natural.

Why is there no subject eu in quero? Should it be Eu quero?

Portuguese is a pro‑drop language: subject pronouns are often omitted because the verb ending already tells you who the subject is.

Quero already indicates eu (I) from its ending ‑o.
Eu quero is also correct, but adding eu usually gives emphasis, like “I want…”.

So:

Hoje à noite quero fazer… – neutral, normal.
Hoje à noite eu quero fazer… – emphasizes that I want to do it (contrast, insistence, etc.).

Why is it quero fazer and not quero a fazer or something with a preposition?

In Portuguese, many verbs of desire, plan, or intention take another verb directly in the infinitive, with no preposition:

quero fazer – I want to do
posso fazer – I can do
vou fazer – I am going to do

So quero fazer is the standard structure:
quero (I want) + fazer (to do / to make) directly.

Saying quero a fazer is incorrect in this meaning.

What is the nuance of fazer um passeio? Could I also say dar um passeio?

Both are used and understood in European Portuguese:

fazer um passeio – to go for a walk / to take a stroll / to go for an outing
dar um passeio – very common; same basic meaning

Nuance:

dar um passeio is very idiomatic and frequent.
fazer um passeio is also correct and common; some speakers might feel it’s slightly less colloquial or just a variation.

In your sentence, quero fazer um passeio longo and quero dar um passeio longo are both natural.

Why is the adjective after the noun in um passeio longo? Could I say um longo passeio?

In Portuguese, the default position of most adjectives is after the noun:

um passeio longo – a long walk

You can say um longo passeio, and it is correct, but:

um passeio longo – neutral, descriptive.
um longo passeio – can sound a bit more literary or emphatic, adding a slight stylistic flavor.

Everyday speech normally uses um passeio longo here.

In depois do jantar, what is do? Why not depois de o jantar?

Do is a contraction of:

de (of / from / after, in this context)
o (the, masculine singular)

So: de + o → do

Literally: depois do jantar = after the dinner.
You wouldn’t write depois de o jantar in normal modern Portuguese; it’s always contracted to depois do jantar.

Could I say depois de jantar instead of depois do jantar? What’s the difference?

Yes, both are correct, but there is a slight nuance:

depois do jantar – after the dinner / after dinner (the meal)
- jantar here is a noun.
depois de jantar – after having dinner / after eating dinner
- jantar here is a verb in the infinitive.

In practice, they often overlap.
Your sentence with depois de jantar is also natural:
Hoje à noite quero fazer um passeio longo depois de jantar.
It sounds a bit more like “after I have dinner” than “after dinner (as an event)”.

Is jantar here a noun or a verb?

In depois do jantar, jantar is a noun meaning dinner:

article o (the) + noun jantar → o jantar

So the structure is: depois do jantar = after the dinner.

If you said depois de jantar, then jantar would be a verb in the infinitive: to have dinner / to dine.

If I change the word order, is Quero fazer um passeio longo hoje à noite depois do jantar still correct?

Yes, that word order is perfectly correct and natural.

Portuguese allows some flexibility with adverbial phrases like hoje à noite and depois do jantar:

Hoje à noite quero fazer um passeio longo depois do jantar.
Quero fazer um passeio longo hoje à noite depois do jantar.
Quero, depois do jantar, fazer um passeio longo hoje à noite. (more marked, with commas)

All are grammatically fine; the differences are mostly about rhythm and emphasis, not correctness.