Je nachdem, wie das Wetter ist, gehen wir in den Park oder bleiben zu Hause.

Je nachdem means depending on that / depending on how things are. In natural English, this sentence starts with something like Depending on what the weather is like...

It often introduces a condition that can change the result:

• Je nachdem, wie spät es ist, ... = Depending on what time it is, ...
• Je nachdem, ob er kommt, ... = Depending on whether he comes, ...

So here, the idea is: the plan changes according to the weather.

Because wie das Wetter ist is an embedded question inside a larger sentence.

In German:

• Direct question: Wie ist das Wetter?
• Embedded question: ..., wie das Wetter ist.

In embedded questions, the conjugated verb usually goes to the end of the clause. That is why ist comes last.

Literally, it is how the weather is, but in natural English we usually say what the weather is like.

So:

• wie das Wetter ist = what the weather is like

German uses wie here where English often uses what ... like.

There are commas because the opening part is not a simple main clause; it contains a subordinate structure.

Two things are happening:

1. Je nachdem introduces the idea of depending on...
2. wie das Wetter ist is an embedded clause

So German sets that whole introductory part off with commas:

• Je nachdem, wie das Wetter ist, gehen wir ...

This is very normal German punctuation.

Because the sentence begins with an introductory clause:

• Je nachdem, wie das Wetter ist, = first position in the sentence

After that, the main clause follows, and in a German main clause the conjugated verb must come in second position. Since the first position is already taken by the introductory clause, the verb comes next:

• Je nachdem, wie das Wetter ist, gehen wir in den Park.

This is the usual verb-second pattern in German.

Because the subject is the same in both parts: wir.

German often leaves out a repeated subject in coordinated clauses when it is clear from context:

• ... gehen wir in den Park oder bleiben zu Hause.

This is understood as:

• ... gehen wir in den Park oder wir bleiben zu Hause.

Both are possible, but leaving out the second wir is very natural.

Because gehen in den Park expresses movement toward a destination.

German uses:

• in + accusative for motion into/toward something
• in + dative for location inside something

So:

• in den Park gehen = to go into/to the park
• im Park sein = to be in the park

Here, we are talking about going there, so in den Park is correct.

Because bleiben describes a location, not movement.

• zu Hause = at home
• nach Hause = home / to home with movement

Compare:

• Wir bleiben zu Hause. = We stay at home.
• Wir gehen nach Hause. = We go home.

So with bleiben, zu Hause is the right choice.

They are related, but they are used differently.

• je nachdem is usually followed by a clause
  • Je nachdem, wie das Wetter ist, ...
• je nach is followed by a noun
  • Je nach Wetter gehen wir in den Park oder bleiben zu Hause.

So both can work here, but the structure changes:

• Je nachdem, wie das Wetter ist, ...
• Je nach dem Wetter ...

The first is a little more explicit because it uses a full clause.

Yes, it is understood as a full coordinated clause, even though the subject is omitted.

Full version:

• ... oder wir bleiben zu Hause.

Shorter version:

• ... oder bleiben zu Hause.

German often shortens repeated material when it is obvious. The meaning stays the same.

Yes. A few natural alternatives are:

• Je nach Wetter gehen wir in den Park oder bleiben zu Hause.
• Wenn das Wetter gut ist, gehen wir in den Park, sonst bleiben wir zu Hause.

The original sentence is very natural, but these versions focus slightly differently:

• je nachdem / je nach = depending on
• wenn ... sonst ... = if ... otherwise ...

So the given sentence is standard and idiomatic German.