Wir grillen im Garten.

Because the subject is wir (we). In the present tense, the verb grillen is conjugated like this:

• ich grille — I grill / I am grilling
• du grillst — you grill
• er/sie/es grillt — he/she/it grills
• wir grillen — we grill
• ihr grillt — you all grill
• sie/Sie grillen — they grill / you (formal) grill

So wir grillen is the correct form for we.

It can mean both.

German does not usually have a separate everyday form for we are grilling the way English does. The normal present tense often covers both ideas:

• Wir grillen. = We grill / We are grilling

The exact meaning depends on context.
For example:

• Jeden Sonntag grillen wir. = We grill every Sunday.
• Gerade grillen wir im Garten. = We’re grilling in the garden right now.

Im is a contraction of in dem.

• in dem Garten → im Garten

This is very common in German, and in most normal sentences im sounds much more natural than the full in dem.

So:

• im Garten = in the garden

Because this sentence describes location, not movement.

German uses in with two different cases:

• dative for location: Where?
• accusative for direction/movement: Where to?

So:

• Wir grillen im Garten. = We are in the garden.
• Wir gehen in den Garten. = We go into the garden.

Here, nobody is moving into the garden. The sentence just says where the grilling happens, so dative is used: im Garten.

Garten is a masculine noun:

• der Garten — the garden

After in in a location meaning, it takes the dative case:

• dem Garten

And because in dem contracts to im, you get:

• im Garten

So the hidden structure is:

• Wir grillen in dem Garten.

Because all German nouns are capitalized.

So:

• wir is not capitalized here because it is a pronoun in the middle of the sentence
• Garten is capitalized because it is a noun

This is a basic rule in German spelling.

Yes. That is completely normal.

In this sentence, grillen works a bit like to barbecue in English: the food is understood from the context.

• Wir grillen im Garten. = We’re grilling / having a barbecue in the garden.

If you want, you can also name the food:

• Wir grillen Würstchen im Garten. — We’re grilling sausages in the garden.
• Wir grillen Gemüse im Garten. — We’re grilling vegetables in the garden.

Yes, but the verb still has to stay in the second position in a normal statement.

The neutral order is:

• Wir grillen im Garten.

You can also move im Garten to the front for emphasis:

• Im Garten grillen wir.

That might sound like:

• We’re grilling in the garden
  or
• It’s in the garden that we’re grilling

So the order can change, but the conjugated verb grillen must still come second.

Not necessarily.

In German, Garten can refer to the outdoor area around a house, which in English might sometimes be called a yard or backyard, depending on the situation.

So im Garten often means:

• in the garden
• out in the yard
• outside in the back garden/backyard

The exact English word depends on context, but Garten is the normal German word.