Outubro é o mês de que eu me lembro quando penso no começo da primavera.

Because it is the first word of the sentence.

In Portuguese, names of months are normally written in lowercase: outubro, novembro, dezembro.
But at the beginning of a sentence, they are capitalized just like in English.

So:

• Outubro é o mês... = capitalized because it starts the sentence
• Eu gosto de outubro. = lowercase in the middle of a sentence

Portuguese usually uses the definite article more often than English.

So o mês literally means the month, and in a sentence like this it sounds natural to say:

• Outubro é o mês...

Leaving out o would sound incomplete or unnatural here.

This is very common in Portuguese, where nouns often appear with articles even when English would not emphasize them as much.

Because the verb expression is lembrar-se de something.

So if you say:

• Eu me lembro de outubro = I remember October

then in a relative clause the de has to stay:

• o mês de que eu me lembro = the month that I remember / the month I remember

This is a very important point in Portuguese: if a verb requires a preposition, that preposition usually stays before the relative pronoun.

Compare:

• a pessoa de quem eu gosto = the person that I like
  because gostar de
• a cidade em que moro = the city where I live
  because morar em

Me lembro comes from the pronominal verb lembrar-se.

• eu me lembro = I remember
• você se lembra = you remember
• nós nos lembramos = we remember

That me is a reflexive/clitic pronoun that belongs to the verb form.

In Brazilian Portuguese, you will hear both patterns:

• lembrar-se de algo
• lembrar algo

But lembrar-se de is very common and often taught as the standard structure in sentences like this.

So here:

• eu me lembro de outubro

becomes:

• o mês de que eu me lembro

Yes. De que and do qual can both work here.

• o mês de que eu me lembro
• o mês do qual eu me lembro

Both are grammatically correct.

The difference is mostly style:

• de que is very common and natural
• do qual is a bit more formal or heavier in style

In everyday speech, many Brazilians would probably prefer de que.

Because Portuguese often omits subject pronouns when the verb form already makes the subject clear.

So:

• eu me lembro
• quando penso

Both refer to I.

The speaker could also say:

• quando eu penso

But Portuguese often drops eu if it is obvious from the verb ending.

Here, penso already clearly means I think, so eu is unnecessary.

Both are correct.

• quando penso no começo da primavera
• quando eu penso no começo da primavera

The version without eu is a little more concise and very natural in Portuguese.

Adding eu can give slight emphasis or simply make the sentence a bit more explicit, but in this context there is no major meaning difference.

No is a contraction of em + o.

So:

• em o começo → no começo

This is very common in Portuguese. Some important contractions are:

• em + o = no
• em + a = na
• em + os = nos
• em + as = nas

So:

• penso no começo = I think about the beginning / start

Da is a contraction of de + a.

So:

• de a primavera → da primavera

This also happens all the time in Portuguese:

• de + o = do
• de + a = da
• de + os = dos
• de + as = das

Here:

• o começo da primavera = the beginning of spring

Literally, the beginning of the spring, though in natural English we usually just say the beginning of spring.

Because the verb pensar commonly uses em when it means to think about something.

So:

• pensar em algo = to think about something

That is why the sentence has:

• penso no começo da primavera

since no = em + o

Using sobre is possible in some contexts, but it often sounds more like about in the sense of discussing a topic. For ordinary mental association, pensar em is the usual choice.

Yes, in this sentence they are very close in meaning.

• o começo da primavera
• o início da primavera

Both mean the beginning/start of spring.

Começo is often slightly more everyday and conversational.
Início can sound a little more formal or neutral depending on context.

In this sentence, either would work naturally.

In Portuguese, seasons are normally written in lowercase:

• primavera
• verão
• outono
• inverno

That is different from English, where learners sometimes wonder whether they should capitalize them. In Portuguese, you usually do not capitalize them unless they begin the sentence or are part of a title.

Yes, it is natural and grammatically correct.

The structure is:

• Outubro = subject
• é o mês = is the month
• de que eu me lembro = that I remember
• quando penso no começo da primavera = when I think about the beginning of spring

It is a slightly thoughtful or literary way to express an association, but it is still perfectly natural Portuguese.

A more conversational rewording might be something like:

• Quando penso no começo da primavera, o mês de que eu me lembro é outubro.

But the original sentence is completely fine.

In careful standard grammar, the expected form is:

• o mês de que eu me lembro

because lembrar-se requires de.

In everyday speech, some speakers may simplify structures and drop prepositions in relative clauses, but that is not the form usually taught to learners as the standard.

So if you are learning correct, reliable Portuguese, use:

• de que eu me lembro
• or do qual eu me lembro

Because Portuguese grammar often keeps the preposition required by the verb or noun, and then contracts that preposition with articles.

In this one sentence you have several examples:

• de que because of lembrar-se de
• no = em + o
• da = de + a

This is one of the biggest differences from English. English often drops or moves prepositions in ways Portuguese does not.

So a useful learning strategy is:

1. learn the verb with its preposition
   
   • lembrar-se de
   • pensar em
2. notice article contractions
   
   • em + o = no
   • de + a = da

That makes sentences like this much easier to understand.