Nie wiem, czy jutro jedziemy w góry czy zostajemy w domu.

The two czy do two slightly different jobs.

• The first czy means whether/if and introduces an indirect question: Nie wiem, czy... = I don't know whether...
• The second czy introduces the alternative: ...czy zostajemy w domu = ...or we stay home

So the whole pattern is basically I don’t know whether X or Y.

Because czy is used for uncertainty about a yes/no question or a choice, while że introduces a statement.

Compare:

• Wiem, że jutro jedziemy w góry. = I know that we’re going to the mountains tomorrow.
• Nie wiem, czy jutro jedziemy w góry. = I don’t know whether we’re going to the mountains tomorrow.

So after nie wiem you use czy when the content is uncertain.

In Polish, present-tense forms of imperfective verbs are often used for the future when the context already makes the time clear.

Here:

• jedziemy literally looks like we are going / we go
• zostajemy literally looks like we are staying / we stay

But because of jutro (tomorrow), both are understood as future. This is very similar to English sentences like Tomorrow we’re going to the mountains.

This is about movement vs. location.

• w góry means movement to the mountains
• w domu means being/staying in/at home

So:

• with movement, Polish often uses w + accusative
• with location, Polish uses w + locative

Here, jedziemy w góry means we’re going to the mountains, while zostajemy w domu means we’re staying at home.

Because Polish normally uses the plural góry for the mountains as a destination or region.

• jechać w góry = to go to the mountains
• góra = a mountain / hill (singular)

So w góry is the normal expression here. Be careful not to confuse it with w górę, which means upwards.

Because zostajemy w domu describes location, not movement.

• w domu = at home / in the house
• do domu = to home / homeward, used with verbs of motion

So:

• Zostajemy w domu. = We’re staying home.
• Jedziemy do domu. = We’re going home.

The comma before the first czy is definitely needed, because czy introduces a subordinate clause after Nie wiem.

You will also often see a comma before the second czy in careful writing:

Nie wiem, czy jutro jedziemy w góry, czy zostajemy w domu.

That version is very common in standard written Polish, because it clearly separates the two alternatives.

Usually, in this kind of sentence, repeating czy is the most natural choice.

Polish often uses the pattern:

czy X, czy Y = whether X or Y

Using albo is possible in other kinds of coordination, but here czy... czy... sounds more natural and idiomatic.

Yes. Polish word order is fairly flexible.

For example, these are also natural:

• Nie wiem, czy jedziemy jutro w góry, czy zostajemy w domu.
• Nie wiem, czy jutro w góry jedziemy, czy zostajemy w domu.
  (more marked, with emphasis)

The basic meaning stays the same, but the emphasis shifts slightly. In your original sentence, jutro naturally sets the time early in the clause.

This is a difference between imperfective and perfective verbs.

• jedziemy, zostajemy = imperfective forms, often used for plans or arrangements
• pojedziemy, zostaniemy = perfective future forms, often focusing more on the completed choice/event

So both of these can work:

• Nie wiem, czy jutro jedziemy w góry, czy zostajemy w domu.
• Nie wiem, czy jutro pojedziemy w góry, czy zostaniemy w domu.

The first can sound a bit like talking about tomorrow’s plan or arrangement. The second can sound a bit more like a single future decision or outcome.

Because zostawać / zostać is the normal everyday verb for stay in this context.

• zostajemy w domu = ordinary, natural, conversational Polish
• pozostajemy w domu = more formal, bookish, or stylistically heavier

So a learner should normally use zostajemy w domu.