Одним из первых симптомов была странная слабость, а потом начался кашель.

Because одним из первых симптомов is not the grammatical subject of the sentence. The subject is странная слабость.

A more neutral word order would be:

Странная слабость была одним из первых симптомов.

In that version, it is easier to see the structure:

• странная слабость = subject, nominative
• одним из первых симптомов = predicate noun phrase, instrumental

After быть in the past tense, Russian often uses the instrumental for the noun phrase that means was X, especially when identifying something as one among a group.

So:

• один из первых симптомов = one of the first symptoms as a standalone phrase
• была одним из первых симптомов = was one of the first symptoms inside the sentence

Because of the pattern один из / одна из / одно из / одни из, which means one of.

After из, Russian uses the genitive case. So:

• симптомы = nominative plural
• из симптомов = of the symptoms / from the symptoms
• из первых симптомов = of the first symptoms

That is why you get:

• одним из первых симптомов

The adjective первых also matches симптомов, so it is genitive plural too.

Because it agrees with слабость, which is the grammatical subject.

• слабость is feminine singular
• so the past tense of быть is была

Even though the sentence starts with Одним из первых симптомов, that phrase is not the subject. Russian word order is flexible, so the subject can come later.

So the core structure is really:

• слабость была ...
• not симптомов было ...

It is not strange, but it is less neutral than the most straightforward order.

Neutral order:

• Странная слабость была одним из первых симптомов, а потом начался кашель.

Actual sentence:

• Одним из первых симптомов была странная слабость, а потом начался кашель.

By putting Одним из первых симптомов first, the speaker gives that idea more emphasis. It sounds a bit like:

• One of the first symptoms was...
• rather than
• A strange weakness was one of the first symptoms...

So the word order helps guide attention.

Because the verb here is начаться, not начать.

• начать = to start something, usually transitive
  
  • Он начал разговор = He started the conversation
• начаться = to begin / to start, when the event itself begins
  
  • Начался дождь = It started raining / The rain began
  • Начался кашель = A cough began

So кашель is the subject of начался.

This is a very common Russian pattern:

• началась боль
• началась паника
• начался дождь
• начался кашель

The -ся marks the verb as reflexive, but in many cases that does not mean a literal English-style reflexive meaning like himself or itself.

With начаться, the reflexive form has become the normal way to say that something begins on its own:

• Концерт начался. = The concert began.
• Кашель начался. = The cough began.

So here -ся is just part of the standard verb начаться.

А потом means and then / after that, but а often adds a slight sense of transition or contrast.

Here the idea is:

• first there was one symptom
• then another thing followed

So а потом works very naturally as and then afterward or and after that.

Compare:

• и потом = more simply and then
• а потом = often feels like and then, after that, at the next stage

In this sentence, а потом sounds very natural because it moves the story forward.

Слабость usually means weakness.

In a medical context, it often means general physical weakness, low energy, or a feeling of being weak. So here странная слабость suggests something like:

• unusual weakness
• a strange feeling of weakness
• unexplained weakness

It does not usually mean weakness in the moral sense here. The medical context makes the physical meaning clear.

Yes. That is completely normal in Russian.

Russian often treats symptoms and conditions as things that begin:

• начался кашель = a cough began
• началась температура = a fever started
• началась боль = pain started

This may sound slightly different from the most natural English phrasing, but in Russian it is very idiomatic.

Yes, absolutely.

That version is more neutral and straightforward. It presents the subject first:

• Странная слабость была одним из первых симптомов.

The original version:

• Одним из первых симптомов была странная слабость.

puts more focus on one of the first symptoms and then tells you what it was.

Both are correct. The difference is mainly emphasis and information flow.

Because а is a coordinating conjunction, and here it joins two clauses:

1. Одним из первых симптомов была странная слабость
2. потом начался кашель

Russian normally puts a comma before а in this kind of sentence.

So the comma is standard punctuation, not something unusual about this example.