Październik bywa zimny, ale nadal lubię spacerować po parku.

Yes—months are normally written with a lowercase letter in Polish: styczeń, luty, październik.

Here it is capitalized only because it is the first word of the sentence. So inside a sentence, you would normally write:

Lubię październik.

not

Lubię Październik.

Bywa is a very useful word in Polish. It often means something like:

• can be
• is sometimes
• tends to be from time to time

So:

• Październik jest zimny = October is cold
  This sounds more direct and general.
• Październik bywa zimny = October can be cold / October is sometimes cold
  This sounds less absolute.

So bywa suggests that October is not always cold, but it often is, or it can be.

Bywa is the 3rd person singular present tense of bywać.

The verb bywać is related to być (to be), but it has a more habitual/repeated/occasional sense.

So:

• jest = is
• bywa = is sometimes / tends to be / can be

Because październik is singular, we use bywa, not a plural form.

Because październik is a noun, and zimny is an adjective describing it.

• październik = masculine singular noun
• zimny = masculine singular adjective

They must agree:

• Październik bywa zimny.

But zimno is usually used differently, for example in impersonal weather expressions:

• Jest zimno. = It is cold.

So:

• Październik jest zimny. = adjective describing a noun
• Jest zimno. = impersonal statement about the weather/temperature

In Polish, ale (but) normally introduces a new clause, and it is preceded by a comma.

So this is standard punctuation:

Październik bywa zimny, ale nadal lubię spacerować po parku.

This works much like English:

October can be cold, but I still like walking in the park.

Yes, here nadal means still.

So:

ale nadal lubię... = but I still like...

It shows contrast:

• October can be cold
• but despite that, I still like walking in the park

Other common Polish words that can sometimes mean still include wciąż and ciągle, but nadal is very natural here.

After lubić (to like), Polish very often uses an infinitive to express liking an activity:

• Lubię czytać. = I like reading / I like to read.
• Lubię pływać. = I like swimming / I like to swim.
• Lubię spacerować. = I like walking / I like to walk.

So lubię spacerować is the most natural pattern.

A noun like spacerowanie exists, but it is usually less natural in ordinary speech here.

Spacerować is the imperfective verb, and that is exactly what Polish normally uses for:

• general habits
• repeated actions
• activities in progress
• things you like doing in general

Here the sentence means that the speaker generally likes walking in the park, not that they completed one specific walk.

That is why spacerować is correct.

Both can relate to the park, but they give slightly different ideas.

spacerować po parku

This means something like:

• walk around the park
• walk through the park
• stroll about in the park

It emphasizes moving around within that space.

spacerować w parku

This means more literally:

• walk in the park

It is understandable, but po parku is especially natural with spacerować, because strolling usually suggests moving around the area.

So in this sentence, po parku is a very good, idiomatic choice.

Parku is the locative singular of park.

That happens because the preposition po takes the locative case when it means movement around an area.

So:

• park = nominative
• po parku = locative after po

This is a very common pattern:

• chodzić po mieście = walk around the city
• biegać po lesie = run around the forest
• spacerować po parku = walk around the park

Lubię is 1st person singular present tense of lubić.

It means:

• I like

So the speaker is saying:

I still like walking around the park.

Some present-tense forms of lubić are:

• lubię = I like
• lubisz = you like
• lubi = he/she/it likes
• lubimy = we like
• lubicie = you (plural) like
• lubią = they like

Yes, Polish word order is fairly flexible, although some versions sound more natural or put emphasis in different places.

The original:

Październik bywa zimny, ale nadal lubię spacerować po parku.

You could also say:

• Październik bywa zimny, ale lubię nadal spacerować po parku.
  Grammatically possible, but less natural.
• Nadal lubię spacerować po parku, chociaż październik bywa zimny.
  Very natural, with a slightly different structure.

The original sentence sounds smooth and neutral.

A rough English-friendly pronunciation is:

PAZH-dzhyer-neek

A few important points:

• ź is a soft sound, somewhat like a softened zh
• dz is pronounced together
• ie often sounds like ye
• stress in Polish is usually on the second-to-last syllable, so here it falls on -dzier-

So the stress is roughly:

paź-DZIER-nik

Yes, it sounds natural and idiomatic.

It has a very normal structure:

• a general observation: Październik bywa zimny
• followed by contrast: ale nadal lubię spacerować po parku

It sounds like something a native speaker could easily say in everyday conversation or writing.