Learn Hardcore Portuguese (Portugal): Eu preciso de companhia calma quando estou triste; a solidão aumenta a minha fraqueza emocional. - I need calm company when I am sad; being alone increases my emotional weakness.

You should think of precisar de as a fixed pattern, like to depend on in English.
Saying eu preciso companhia (without de) is ungrammatical in European Portuguese and sounds wrong to most speakers.

Could you say eu preciso de uma companhia calma instead of eu preciso de companhia calma?

Yes, you can. Both sound natural, but there is a small nuance:

preciso de companhia calma – more general, “I need calm company” as an abstract idea.
preciso de uma companhia calma – a bit more concrete, as if you’re talking about a kind/instance of company that is calm (for example, one calm person to be with you).

In everyday speech, both are possible; the version without uma feels slightly more generic and stylized.

Is the eu in eu preciso de companhia calma necessary, or could it be left out?

It can absolutely be left out:

Preciso de companhia calma…

Portuguese usually drops subject pronouns because the verb ending (preciso) already shows the person (1st person singular).
Using eu adds a bit more emphasis on I:

Eu preciso de companhia calma (maybe contrasting with others, or stressing that this is my need).

Why is it quando estou triste and not quando eu estou triste?

Both are correct:

quando estou triste
quando eu estou triste

Again, Portuguese often omits subject pronouns, so eu is optional.
Including eu adds emphasis or contrast:

Quando eu estou triste, preciso de companhia calma. (Maybe implying: that’s my case, even if others are different.)

Without eu, it just sounds a bit smoother and more neutral.

What is the difference between estou triste and sou triste?

estou triste – temporary state, I am feeling sad (now / these days).
sou triste – permanent characteristic, I am a sad person by nature.

In your sentence you clearly mean a temporary emotion, so estou triste is the correct choice. Sou triste would sound like a personality description and is much stronger.

Why is it companhia calma instead of calma companhia?

In Portuguese, adjectives usually come after the noun:

companhia calma – normal word order.

Putting the adjective before the noun (calma companhia) is possible but much more literary or poetic, and it often adds a special nuance or emphasis.

So in normal speech and writing, companhia calma is the natural order.

Why do both words companhia and calma end in -a?

Because of gender agreement:

companhia is a feminine noun (it ends in -a, and its article is a companhia).
calma is a feminine singular adjective, agreeing with companhia.

In Portuguese, adjectives must agree in gender and number with the noun:

companhia calma (feminine singular)
amigo calmo (masculine singular)
amigas calmas (feminine plural), etc.

Why is there a semicolon (;) instead of a comma or a period?

The semicolon is a stylistic choice. It separates two closely related independent clauses:

Eu preciso de companhia calma quando estou triste; a solidão aumenta a minha fraqueza emocional.

You could also write:

with a period:
…quando estou triste. A solidão aumenta…
with a comma: less correct in formal writing, because both parts are full sentences.

The semicolon nicely shows that the second idea explains or develops the first one.

Why is it a solidão and not just solidão?

In Portuguese, abstract nouns often use the definite article when you talk about them in a general sense:

a solidão = loneliness in general
a vida, o amor, a tristeza, etc.

So a solidão aumenta… means loneliness (as a general condition) increases…
Leaving out the article (solidão aumenta…) is possible but sounds very clipped, more like a title or a note than a normal sentence.

Why is it a minha fraqueza emocional instead of minha fraqueza emocional?

In European Portuguese, possessives almost always come with a definite article:

a minha fraqueza, o meu carro, as minhas amigas.

So a minha fraqueza emocional is the natural European form.
Dropping the article (minha fraqueza emocional) is much more common in Brazilian Portuguese and in poetry or very stylized language. In everyday European Portuguese, it sounds foreign or marked.

Why is the adjective emocional after fraqueza and not before, like in English?

Portuguese usually puts descriptive adjectives after the noun:

fraqueza emocional = emotional weakness
problema grave = serious problem
vida tranquila = calm life

Some adjectives can come before the noun with a slightly different nuance, but emocional is typically placed after.
So fraqueza emocional is the natural order; emocional fraqueza would sound odd or archaic.

Why is the verb aumenta in the 3rd person singular?

Because it agrees with its subject, a solidão:

a solidão → 3rd person singular
verb aumentar in present: (ela) aumenta

So:

A solidão aumenta a minha fraqueza emocional.
Loneliness increases my emotional weakness.

If the subject were plural, the verb would also be plural:

As dificuldades aumentam a minha fraqueza emocional.

Could I say quando fico triste instead of quando estou triste? Is there any difference?

Yes, you can say both:

quando estou triste – focuses on the state: when I am sad.
quando fico triste – focuses on the change: when I become / get sad.

The meaning is very similar, but fico triste highlights the moment of becoming sad a bit more. In your sentence, either option would be natural.