Я и зимой люблю гулять в парке.

Not in the basic and sense.

In Я и зимой люблю гулять в парке, и means something like:

• also
• too
• sometimes even

So и зимой means in winter too or even in winter.

The exact nuance depends on context:

• Я люблю гулять в парке летом. Я и зимой люблю гулять в парке.
  = I like walking in the park in summer. I like it in winter too.

• If the speaker is stressing that winter is surprising, it can feel like:
  I even like walking in the park in winter.

Because Russian often uses the instrumental case for seasons when talking about time.

So:

• зима = winter
• зимой = in winter

This is a very common pattern:

• весной = in spring
• летом = in summer
• осенью = in autumn
• зимой = in winter

Here зимой is functioning almost like an adverb of time.

Normally, no.

For the season meaning, Russian usually says:

• зимой = in winter

в зиме is not the normal way to say in winter. It would sound wrong in this context.

You may sometimes see other expressions, such as:

• в эту зиму = this winter
• в зимнее время = in the wintertime

But for the simple general idea, зимой is the standard choice.

Because after люблю when you mean I like doing something, Russian uses the infinitive.

So:

• люблю гулять = I like to walk / I like walking
• люблю читать = I like to read / I like reading
• люблю плавать = I like to swim / I like swimming

So гулять stays in the infinitive because it is the action that is liked.

Гулять usually means to walk, to take a walk, or to stroll, often for pleasure, relaxation, or fresh air.

So here it is not just movement from one place to another. It suggests something more like:

• going for a walk
• walking around leisurely
• spending time outdoors

That is why гулять в парке sounds very natural: parks are classic places for гулять.

Because the meaning is different.

• идти = to be going, to walk in one direction right now
• ходить = to go/walk regularly or back and forth
• гулять = to walk for pleasure, stroll, spend time walking outside

Compare:

• Я иду в парк. = I’m going to the park.
• Я хожу в парк. = I go to the park regularly.
• Я люблю гулять в парке. = I like walking/strolling in the park.

So гулять is the best choice if the idea is enjoying walks there.

Because в парке means in the park, a location.

Russian uses different cases after в depending on meaning:

• в парк = into the park, to the park
  motion/direction
• в парке = in the park
  location

Here the sentence is about where the walking happens, so it uses the prepositional case:

• парк → в парке

Yes, Russian word order is flexible, and different orders change the emphasis.

This version highlights и зимой fairly strongly:

• Я и зимой люблю гулять в парке.
  = I like walking in the park in winter too / even in winter.

Other possible versions:

• Я люблю гулять в парке и зимой.
  More neutral; the winter too idea comes at the end.

• Зимой я люблю гулять в парке.
  More like In winter, I like to walk in the park.

So the given order helps draw attention to winter as the important point.

Гулять is imperfective, and that is the normal choice for a general habit or something you like doing in general.

• люблю гулять = I like walking / I like to go for walks

Russian usually prefers the imperfective after любить when talking about repeated or habitual actions.

You might also hear:

• люблю погулять

That is possible, but it has a slightly different feel: more like I like to go for a walk as a complete outing or occasion. In this sentence, гулять is the most neutral and standard choice.

Yes, often it could.

Russian frequently drops subject pronouns when they are clear from the verb:

• И зимой люблю гулять в парке.

This is possible and natural in the right context.

But Я may be kept for:

• clarity
• emphasis
• contrast

For example, if the speaker is contrasting themselves with someone else:

• Он зимой не любит гулять, а я и зимой люблю гулять в парке.
  = He doesn’t like walking in winter, but I do.

So Я is not always necessary, but it can be useful.

It normally describes a general preference or habit.

• люблю = I like
• гулять = to walk / walking
• зимой = in winter

So the natural reading is:

• I like walking in the park in winter too
• Even in winter, I like walking in the park

It is not about one single occasion unless extra context makes it specific.

The main stresses are:

• Я
• и зимой́
• люблю́
• гуля́ть
• в па́рке

A rough pronunciation guide:

• Я = ya
• и зимой́ = ee zee-MOY
• люблю́ = lyu-BLYU
• гуля́ть = goo-LYAT’
• в па́рке = f PAR-ke

The в often sounds like f before the п in парке, because of devoicing.