Im Garten schneide ich die Zucchini, und meine Cousine wäscht die Linsen.

Because German main clauses normally follow the verb-second rule.

That means:

• one element comes first,
• the finite verb comes second.

Here, Im Garten is put first, so the verb schneide must come next:

• Im Garten schneide ich die Zucchini.

If you started with the subject instead, you would get:

• Ich schneide die Zucchini im Garten.

Both are correct, but the original sentence gives extra emphasis to Im Garten.

Im is a contraction of in dem.

So:

• im Garten = in dem Garten = in the garden

German often combines certain prepositions and articles:

• in dem → im
• an dem → am
• zu dem → zum
• zu der → zur

Using im here is the normal, natural choice.

Because this sentence describes a location, not movement.

German uses:

• dative after in for location: im Garten = in the garden
• accusative after in for movement into somewhere: in den Garten = into the garden

Compare:

• Ich bin im Garten. = I am in the garden.
• Ich gehe in den Garten. = I go into the garden.

Here, the speaker is already there, so im Garten is correct.

Because Zucchini is one of those nouns whose singular and plural often look the same.

So:

• die Zucchini can mean the zucchini (singular, feminine)
• die Zucchini can also mean the zucchinis (plural)

The article die does not help here, because:

• feminine singular definite article = die
• plural definite article = die

So only the context tells you whether one or more are meant.

These are just the present-tense forms of two different verbs:

• schneiden → ich schneide
• waschen → meine Cousine wäscht

A few details:

• schneide is the 1st person singular form: I cut / am cutting
• wäscht is the 3rd person singular form: she washes / is washing

Also, waschen is a strong verb, so its stem vowel changes in the du/er/sie/es forms:

• ich wasche
• du wäschst
• er/sie/es wäscht

That is why you see ä in wäscht.

Because each main clause follows the same verb-second rule, but they start differently.

First clause:

• Im Garten = first element
• schneide = second position
• ich = comes after the verb

Second clause:

• meine Cousine = first element
• wäscht = second position

So the word order changes depending on what comes first, but the rule is the same in both clauses.

Because Cousine is a feminine noun, and here it is the subject of the clause.

• Cousine = female cousin
• mein Cousin = male cousin
• meine Cousine = my female cousin

The possessive adjective mein- changes its ending depending on gender, number, and case. With a feminine noun in the nominative, it becomes meine.

Because Linsen is plural, and it is the direct object of wäscht.

• singular: die Linse = the lens / lentil
• plural: die Linsen = the lentils

In the plural, the definite article is die in both nominative and accusative:

• die Linsen can be subject or object, depending on the sentence

Here it is the object because your cousin is doing the action to them:

• meine Cousine wäscht die Linsen

It can mean either, depending on context.

German does not usually have a separate form exactly like the English progressive am cutting or is washing. So:

• ich schneide can mean I cut or I am cutting
• meine Cousine wäscht can mean my cousin washes or my cousin is washing

In this sentence, the natural meaning is probably the ongoing one:

• I am cutting the zucchini, and my cousin is washing the lentils.

Not usually.

In German, when und connects two main clauses, the comma is often omitted:

• Im Garten schneide ich die Zucchini und meine Cousine wäscht die Linsen.

A comma can sometimes be used for clarity or rhythm, but in a simple sentence like this, many people would leave it out.

So:

• without the comma = very common
• with the comma = possible in some contexts, but less basic/default

Because in German, all nouns are capitalized.

So:

• der Garten
• die Zucchini
• die Cousine
• die Linsen

This is one of the most noticeable differences from English spelling. Adjectives, verbs, and most other words are not capitalized unless they begin a sentence or follow another capitalization rule.

Yes, absolutely.

That version is also correct:

• Ich schneide die Zucchini im Garten.

The difference is mostly about emphasis:

• Im Garten schneide ich die Zucchini emphasizes where
• Ich schneide die Zucchini im Garten is more neutral and starts with the subject

German word order is more flexible than English, but the finite verb still stays in second position in a normal main clause.