Learn Hardcore French: Je ne peux pas supporter le bruit quand je travaille.

Yes. Ne pas supporter already means can’t stand. Adding pouvoir (as in Je ne peux pas supporter) can emphasize inability or make it sound a bit heavier, but both are correct and common.

Why does the negation go around peux (the conjugated verb) and not around supporter?

In multi-verb constructions, ne … pas encloses the conjugated verb: Je ne peux pas + infinitive. So: Je ne peux pas supporter, not Je ne peux supporter pas.

Is it okay in speech to drop ne and say Je peux pas supporter le bruit…?

Yes, very common in informal spoken French. In writing or formal speech, keep ne.

Why is it le bruit here and not du bruit?

Le bruit refers to noise in general (generic use of the definite article). Du bruit means some noise, a quantity. Both are possible, but the original sentence states a general dislike of noise.

What’s the difference between le bruit, des bruits, and du bruit?

le bruit: noise in general (the concept).
du bruit: some noise, an unspecified amount.
des bruits: specific individual noises/sounds (plural).

Could I say Je ne supporte pas de bruit quand je travaille?

Yes. Pas de bruit means no noise (at all). It’s a bit stronger and focuses on the quantity: you don’t tolerate any noise while working.

Can I replace quand with lorsque or pendant que?

Lorsque: near-synonym of quand, slightly more formal/literary.
Pendant que: means “while”; also fine here: … pendant que je travaille.

Is the simple present je travaille the right tense for “when I’m working”? French doesn’t need a continuous form?

Correct. French uses the simple present for both habitual and current actions in a quand clause. If you want to stress “in the middle of,” you can say quand je suis en train de travailler.

How do I pronounce the sentence? Any tricky sounds?

IPA: [ʒə nə pø pa sy.pɔʁ.te lə bʁɥi kɑ̃ ʒə tʁa.vaj]. Tips:

peux [pø] (no x sound).
bruit [bʁɥi] with the [ɥ] glide (like “wee” but with rounded lips).
quand [kɑ̃] nasal vowel.
travaille ends with [aj], not an English “-eye-l” sound. No required liaisons here.

How is pouvoir conjugated here, and how do I pronounce peux vs peut?

Present: je peux, tu peux, il/elle/on peut, nous pouvons, vous pouvez, ils/elles peuvent.
Peux and peut sound the same: [pø].

Is supporter a false friend with English “to support”?

Often, yes. Supporter commonly means “to tolerate/bear/stand.” For “to support (a person/cause),” French usually uses soutenir. As a noun, un supporter is “a fan.”

If I want to replace le bruit with a pronoun, where does it go?

With a conjugated verb + infinitive, the object pronoun goes before the infinitive:

Affirmative: Je peux le supporter.
Negative: Je ne peux pas le supporter. Using le is fine only if the antecedent is clear.

Can I say Je ne peux pas le supporter to mean “I can’t stand the noise”?

Only if le clearly refers to le bruit mentioned earlier. Without context, le could be taken as “him/it.” If there’s no clear antecedent, use ça: Je ne peux pas supporter ça.

Can I move quand je travaille to the front?

Yes: Quand je travaille, je ne peux pas supporter le bruit. Add a comma after the initial clause.

Why not pendant je travaille?

Because pendant is a preposition. Before a clause, use pendant que: … pendant que je travaille.

Are there other natural ways to say this?

Yes, for example:

Je ne supporte pas le bruit quand je travaille.
Le bruit me gêne quand je travaille.
Le bruit m’empêche de travailler.
Je n’arrive pas à travailler quand il y a du bruit.
Je supporte mal le bruit quand je travaille.

What if I want to insist on “no noise at all”?

Use aucun: Je ne supporte aucun bruit quand je travaille. Strong and categorical.

Is travaille correct for je? What’s the difference from travail?

Yes, je travaille is the 1st person singular of travailler (verb). Travail (without -le at the end) is the noun “work.”