Learn Hardcore French: Le bruit de la fontaine le calme après une journée de télétravail. - The sound of the fountain calms him after a day of remote work.

Le is a direct object pronoun meaning him or it. It stands for a masculine singular person or thing mentioned earlier in the context (for example, le voisin, le professeur, le chat).

In French, direct object pronouns usually come before the conjugated verb:

Le bruit de la fontaine le calme. – The sound of the fountain calms him / it.

So the order is: [subject] + [object pronoun] + [verb].
You cannot say Le bruit de la fontaine calme le in standard French.

Is calme here an adjective or a verb? How can I tell?

Here calme is the verb calmer, conjugated in the present tense, 3rd person singular:

infinitive: calmer – to calm (someone/thing)
conjugation: je calme, tu calmes, il/elle calme, nous calmons, vous calmez, ils/elles calment

You can tell it is a verb because:

There is a subject before it: Le bruit de la fontaine.
There is a direct object pronoun before it: le.
The sentence needs a verb to make sense; the only candidate here is calme.

If calme were an adjective meaning calm, you would expect a form of être:

Le bruit de la fontaine est calme. – The sound of the fountain is calm.

Could this sentence mean “The sound of the fountain is calm after a day of remote work”?

No. In this sentence, it cannot mean is calm.

Le bruit de la fontaine le calme… = The sound of the fountain calms him / it… (verb calmer)
To say is calm, you would need est calme:
- Le bruit de la fontaine est calme après une journée de télétravail.

The presence of the object pronoun le forces calme to be read as a verb (calms), not as the adjective calm.

Why is it de la fontaine and not du fontaine?

Because fontaine is a feminine noun: la fontaine.

de + la stays de la → de la fontaine
du is a contraction of de + le and is used only with masculine singular nouns:
- le jardin → du jardin (from / of the garden)
- le chien → du chien

So you get:

le bruit de la fontaine – the sound/noise of the fountain
not le bruit du fontaine, which is ungrammatical.

How do I know that fontaine is feminine?

You mostly have to learn it with the word, just like you learn the plural in English.

Dictionary entries will show une fontaine or n. f. (for féminin).
Many words ending in -aine are feminine (la montagne, la semaine, la cuisine), but this is a tendency, not an absolute rule.

So you memorize: une fontaine, la fontaine → therefore de la fontaine.

What is the difference between une journée and un jour? Could I say après un jour de télétravail?

Both un jour and une journée mean a day, but the nuance is different:

un jour: counts days, more neutral, like a point on a calendar
- He worked for three days → Il a travaillé pendant trois jours.
une journée: emphasizes the duration or what happens during the whole day
- He had a tiring day → Il a eu une journée fatigante.

In this sentence:

après une journée de télétravail stresses the whole day spent working remotely and how long or tiring it was. It sounds very natural and idiomatic.
après un jour de télétravail is grammatically correct, but it sounds more like counting days (day 1, day 2) and is less natural in this context.

Why is it une journée de télétravail and not something like une journée en télétravail?

The pattern une journée de + noun often describes what the day is filled with:

une journée de travail – a day of work
une journée de repos – a day of rest
une journée de pluie – a rainy day (literally a day of rain)

So une journée de télétravail naturally means a day of teleworking / remote work.

You could say en télétravail in other structures (e.g. Je suis en télétravail aujourd’hui – I’m working from home today), but after une journée, de is the usual preposition.

What exactly does télétravail mean, and is it masculine or feminine?

Télétravail means remote work / working from home (literally distance-work).

It is a masculine noun: le télétravail.
In the sentence you have une journée de télétravail – a day of remote work.

Pronunciation (in IPA): [te.le.tʁa.vaj]

té – like tay
lé – like lay
travail – tra-vai (final -l is silent)

If the person being calmed was a woman instead of a man, how would the sentence change?

You would change the direct object pronoun le (him/it, masculine) to la (her/it, feminine):

Le bruit de la fontaine la calme après une journée de télétravail.
– The sound of the fountain calms her after a day of remote work.

The verb calme does not change; it stays the same in the 3rd person singular.

How would I make this sentence negative: “The sound of the fountain does not calm him after a day of remote work”?

You wrap ne … pas around the verb, keeping the object pronoun before the verb:

Le bruit de la fontaine ne le calme pas après une journée de télétravail.

Structure:

ne
- le (object pronoun) + calme (verb) + pas.