Na het kletsen in de tuin gaan wij naar huis.

Because the sentence starts with a different element: Na het kletsen in de tuin.

In Dutch main clauses, when something other than the subject comes first, the finite verb usually takes the second position. This is often called inversion.

• Normal order: Wij gaan naar huis.
• With a fronted time phrase: Na het kletsen in de tuin gaan wij naar huis.

So:

• first position: Na het kletsen in de tuin
• second position: gaan
• then the subject: wij

This is very common in Dutch.

Na het kletsen means after chatting or after the chatting.

Here, kletsen is normally a verb meaning to chat. But after na, Dutch often turns a verb into something like an activity noun by using:

• het + infinitive

So:

• kletsen = to chat
• het kletsen = the chatting / chatting as an activity

This structure is very common:

• na het eten = after eating
• na het werken = after working
• na het zwemmen = after swimming

So na het kletsen is a very natural Dutch way to say after chatting.

Normally, no. In this structure, Dutch usually needs het before the infinitive.

So you say:

• na het kletsen
• na het eten
• na het lezen

Not normally:

• na kletsen
• na eten
• na lezen

The het is part of the standard pattern when an infinitive is used as a noun after a preposition like na.

It most naturally belongs with kletsen:

• Na het kletsen in de tuin = after chatting in the garden

So the idea is that the chatting happened in the garden.

That whole phrase then sets the scene for the main clause:

• gaan wij naar huis = we go home

You could think of the sentence in two parts:

1. Na het kletsen in de tuin
2. gaan wij naar huis

So in de tuin is part of the first part, describing where the chatting took place.

In Dutch, naar huis is the normal expression for homeward / home.

Just like English says go home rather than go to the home, Dutch usually says:

• naar huis gaan = to go home

So huis here works a bit like a fixed adverbial expression, not like a regular noun with an article.

Examples:

• Ik ga naar huis. = I’m going home.
• Wij gaan naar huis. = We’re going home.

But if you mean a specific house, then you can use an article:

• Wij gaan naar het huis van Jan. = We are going to Jan’s house.

So:

• naar huis = home
• naar het huis = to the house

Both mean we.

• wij is the full/stressed form
• we is the unstressed, more common spoken form

So this sentence could also be:

• Na het kletsen in de tuin gaan we naar huis.

That sounds very natural in everyday Dutch.

You often see wij when the subject is being emphasized, contrasted, or written a little more formally. In many normal conversations, we is more common.

Kletsen means to chat, to talk casually, or sometimes to gossip, depending on context.

It is generally informal and often suggests relaxed, friendly conversation.

Examples:

• We zaten te kletsen. = We were chatting.
• Ze kletsen de hele middag. = They chat all afternoon.
• Niet zo veel kletsen! = Don’t talk so much!

In your sentence, it most likely means chatting casually.

Yes. It tells you when the main action happens.

The main action is:

• gaan wij naar huis = we go home

The phrase:

• Na het kletsen in de tuin

means that this happens after the chatting in the garden. So it functions as a time phrase.

Dutch often puts these kinds of time expressions at the beginning of the sentence, which then causes inversion:

• Na het eten ga ik slapen.
• Na het werk gaan we boodschappen doen.

Yes, absolutely.

That version is also grammatical:

• Wij gaan na het kletsen in de tuin naar huis.

The difference is mostly about focus and style.

• Na het kletsen in de tuin gaan wij naar huis.
  This emphasizes the time/sequence first.
• Wij gaan na het kletsen in de tuin naar huis.
  This starts more neutrally with the subject.

Both are correct.

Because the opening phrase is a normal part of the sentence, not a separate clause that needs a comma.

Dutch usually does not require a comma after a simple introductory phrase like this:

• Na het eten gaan we wandelen.
• Morgen ga ik werken.
• In de zomer zwemmen wij vaak.

So:

• Na het kletsen in de tuin gaan wij naar huis.

without a comma is standard.

Yes. Gaan is in the present tense here.

But just like in English, the present tense can often describe something happening now, habitually, or in the near future, depending on context.

So gaan wij naar huis could mean:

• we are going home
• we go home
• we’re going home (next / soon)

In this sentence, because of na het kletsen, it most naturally means something like:

• After chatting in the garden, we go home / we’re going home.

The exact time feeling depends on the wider context.