Какая бы ни была погода, мы всё равно гуляем в парке.

As a whole, Какая бы ни была погода means whatever the weather is like or no matter what the weather is like.

This is a very common Russian pattern for concession:

• что бы ни... = no matter what...
• кто бы ни... = whoever...
• где бы ни... = wherever...
• какой бы ни... = whatever kind of...

So the sentence means:

No matter what the weather is like, we still go for walks in the park.

Because погода is feminine singular, and какая agrees with it.

• погода = feminine
• nominative singular form = погода
• the matching form of какой is какая

So:

• какая погода = what kind of weather
• какая бы ни была погода = whatever the weather is like

It is nominative because погода is the subject of that clause.

This is one of the most common points of confusion.

In Russian, the particle бы combines with a past-tense form to create a conditional or non-literal meaning. So была here does not mean simple past was in the normal sense.

In какая бы ни была погода, the form была is part of a fixed concessive pattern:

• какая бы ни была... = whatever ... is like
• literally something like whatever it might be

So even though была looks past-tense, the whole clause can refer to the present, future, or a general situation.

It agrees with погода, so it is:

• feminine
• singular
• past-form used with бы

Together, бы and ни help create the meaning no matter... / whatever...

Their jobs are different:

• бы adds a conditional or hypothetical sense
• ни strengthens the idea of indefiniteness or concession

So:

• какая погода = what kind of weather
• какая бы ни была погода = whatever the weather is like / no matter what the weather is like

This ...бы ни... pattern is very important in Russian.

Compare:

• Что бы ни случилось, я помогу. = No matter what happens, I will help.
• Кто бы ни пришёл, открой дверь. = Whoever comes, open the door.

Yes, Какая бы погода ни была is also possible.

Both are grammatical, and both mean the same thing:

• Какая бы ни была погода
• Какая бы погода ни была

Russian word order is more flexible than English word order. In this kind of clause, the words can move around somewhat for style, rhythm, or emphasis.

The version in your sentence sounds natural and idiomatic. A learner should mainly recognize the pattern rather than worry too much about the exact word order at first.

Всё равно means all the same, anyway, or still.

In this sentence, it emphasizes that the weather does not change the result:

• Какая бы ни была погода, мы гуляем в парке.
  = Whatever the weather is like, we go for walks in the park.

• Какая бы ни была погода, мы всё равно гуляем в парке.
  = Whatever the weather is like, we still go for walks in the park.

So всё равно adds emphasis. It is very natural here, but the sentence could exist without it.

Гуляем is the 1st person plural present tense of гулять.

Here гулять means to walk, to go for a walk, or to spend time outdoors walking.

Because the sentence expresses a regular or habitual action, Russian uses the imperfective present:

• мы гуляем = we walk / we go for walks

It does not mean one specific completed walk. It describes a general habit.

So the sentence suggests something like:

• We always go walking in the park, regardless of the weather.

Because в + prepositional case is used for being or doing something in a place.

• парк
• prepositional singular = парке
• в парке = in the park

Here the meaning is location:

• гуляем в парке = we walk in the park

If there were movement to the park, Russian would usually use:

• в парк = into the park / to the park

So:

• Мы гуляем в парке. = We are walking / we walk in the park.
• Мы идём в парк. = We are going to the park.

Yes. The comma is necessary.

Какая бы ни была погода is a subordinate concessive clause, and мы всё равно гуляем в парке is the main clause.

Russian normally separates this kind of subordinate clause with a comma, especially when it comes first:

• Какая бы ни была погода, мы всё равно гуляем в парке.

This is similar to English punctuation in:

• Whatever the weather is like, we still go for walks in the park.

Yes. Russian often omits subject pronouns when the verb already makes the subject clear.

So both are possible:

• Какая бы ни была погода, мы всё равно гуляем в парке.
• Какая бы ни была погода, всё равно гуляем в парке.

Because гуляем already means we walk, мы is not strictly necessary.

Including мы can add a little emphasis or clarity, but it is not required.

It is normal, standard Russian. It is not unusually formal.

The pattern какая бы ни была... is a little more structured than very simple everyday speech, but it is absolutely standard and natural. A native speaker would understand it immediately, and it works well in both spoken and written Russian.

A more conversational alternative might be something like:

• В любую погоду мы всё равно гуляем в парке. = We go walking in the park in any weather.
• Несмотря на погоду, мы всё равно гуляем в парке. = Despite the weather, we still walk in the park.

But your original sentence is perfectly natural.