Learn Hardcore Portuguese (Portugal): Eu não gosto de rir de amigos quando estão tristes. - I don't like to laugh at friends when they are sad.

Eu não gosto de… and Não gosto de… both mean I don’t like…
Using Eu adds a bit of emphasis on I (contrast with others), but it’s not required.

Why is não placed before gosto and not somewhere else?

In Portuguese, não normally goes immediately before the main conjugated verb:

Eu não gosto de… = I do not like…
You do not say: Eu gosto não de… (wrong).

If there is an auxiliary verb, não usually goes before the auxiliary:

Eu não estou a rir. = I’m not laughing.

Why is it gosto de rir and not gosto rir?

The verb gostar (to like) almost always takes the preposition de:

gostar de algo – to like something
gostar de fazer algo – to like to do something

So you say:

Eu gosto de rir. = I like to laugh.
Eu não gosto de rir. = I don’t like to laugh.

Leaving out de (gosto rir) is incorrect.

Why is rir in the infinitive, not something like a gerund (rindo)?

After gostar de, Portuguese uses the infinitive:

gostar de rir – to like laughing / to like to laugh
gostar de ler – to like reading / to like to read

The gerund rindo is not used in this structure. Eu não gosto de rindo is wrong.

So Eu não gosto de rir naturally translates as I don’t like laughing / I don’t like to laugh.

In European Portuguese, don’t people usually say rir-se? Should it be rir-se de amigos?

In European Portuguese, rir-se is indeed very common:

rir-se – to laugh (reflexive form)
rir-se de alguém – to laugh at someone

A very natural EP version would be:

Eu não gosto de me rir dos amigos quando estão tristes.

Both rir de and rir-se de exist; in Portugal, many speakers prefer rir-se de in everyday speech. Your sentence with rir de is grammatically fine, though.

What is the difference between rir de amigos and rir com amigos?

rir de amigos = to laugh at friends (they are the target of the laughter, often negative or mocking).
rir com amigos = to laugh with friends (you and your friends are laughing together).

So:

Eu não gosto de rir de amigos quando estão tristes.
→ I don’t like laughing at friends when they’re sad.

would be very different from:

Eu gosto de rir com amigos.
→ I like laughing with friends.

Why is it de amigos and not dos amigos or dos meus amigos?

All are possible, but the meaning changes slightly:

de amigos – a bit more general / indefinite: friends (not strongly specified whose)
dos amigos (de + os) – more specific: the friends (probably a known group)
dos meus amigos – clearly my friends.

In real life, most people would probably say:

Eu não gosto de rir dos meus amigos quando estão tristes.
(I don’t like laughing at my friends when they’re sad.)

Your version is grammatically correct; it just sounds a little more generic.

Why is there no subject pronoun in quando estão tristes? Where is “they”?

In quando estão tristes, the subject is still amigos from the first part of the sentence. Portuguese doesn’t need to repeat the subject pronoun eles:

Eu não gosto de rir de amigos quando estão tristes.
= I don’t like laughing at friends when they are sad.

You can say quando eles estão tristes, but it’s not required, and usually Portuguese drops it if the reference is clear.

Could I say quando eles estão tristes instead of quando estão tristes?

Yes:

Eu não gosto de rir de amigos quando eles estão tristes.

This is correct and perfectly clear. Adding eles can give a tiny bit more emphasis to they, but grammatically both versions are fine. The shorter one is a bit more natural in casual speech.

Why is it estão tristes and not são tristes?

Portuguese uses:

estar for temporary states / conditions
ser for permanent characteristics / identity

Being sad is normally seen as a temporary state, so:

estão tristes = they are (currently) sad.

são tristes would mean something like they are sad people (by nature), which is not the meaning here.

Can I change the word order, like: Quando estão tristes, eu não gosto de rir de amigos?

Yes, that’s fine:

Quando estão tristes, eu não gosto de rir de amigos.

The meaning is the same. Portuguese allows the quando… clause either at the beginning or at the end:

Eu não gosto de rir de amigos quando estão tristes.
Quando estão tristes, eu não gosto de rir de amigos.

Is this sentence natural in European Portuguese, or would natives normally say it differently?

It’s understandable and correct, but a very natural European Portuguese version would usually be something like:

Eu não gosto de me rir dos meus amigos quando eles estão tristes.

Main differences native EP speakers often prefer:

rir-se instead of just rir
dos meus amigos (clear “my friends”)
often adding eles in the second clause is also common, but optional.

Your original sentence is fine as a learner’s sentence; natives might just tweak it for naturalness.