Im Garten wedelt die kleine Hündin noch stärker, wenn sie ihren Ball sieht.

This is because German main clauses follow the V2 rule: the finite verb must be in the second position.

So in:

Im Garten wedelt die kleine Hündin ...

• Im Garten is placed first
• wedelt must come second
• the subject die kleine Hündin comes after the verb

A more neutral order would be:

Die kleine Hündin wedelt im Garten ...

Starting with Im Garten puts a little more focus on the location: in the garden.

Im is a contraction of in dem.

So:

• im Garten = in dem Garten = in the garden

German uses this contraction very often.

Also, in is a two-way preposition:

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

Here it is im Garten because the dog is already there; this is a location, not movement.

There are two things going on here:

1. Hündin means female dog

   • der Hund = dog / male dog
   • die Hündin = female dog

2. die kleine Hündin is in the nominative because it is the subject of the sentence.

The adjective ending in kleine comes from the fact that:

• the noun is feminine
• it is singular
• it has the definite article die
• it is in the nominative

So:

• die kleine Hündin = the little female dog

Because in German, when a dog is the subject, wedeln can naturally mean to wag its tail, even if Schwanz is not stated.

So:

• Die Hündin wedelt can mean the dog is wagging its tail

A fuller version would be:

Die kleine Hündin wedelt mit dem Schwanz.

But that is not always necessary, because the meaning is obvious from context.

Here noch stärker means even more strongly or even more.

Breakdown:

• stark = strong
• stärker = stronger / more strongly
• noch stärker = even stronger / even more strongly

In this sentence, it describes how the dog is wagging, so it works like an adverb in English.

Important: noch does not mean still here.
With a comparative, noch often means even:

• noch größer = even bigger
• noch schneller = even faster
• noch stärker = even more strongly

Wenn often means when or whenever, and sometimes if.

In this sentence:

..., wenn sie ihren Ball sieht.

it means something like when/whenever she sees her ball.

That suggests a repeated or general situation: every time she sees the ball, she wags even more.

A useful comparison:

• wenn = when / whenever / if
• als = when, but only for a single event in the past

So:

• Wenn sie ihren Ball sieht, ... = whenever she sees her ball, ...
• Als sie ihren Ball sah, ... = when she saw her ball, ... one past occasion

Because wenn introduces a subordinate clause, and in German subordinate clauses the conjugated verb usually goes to the end.

So:

• wenn = clause starter
• sie ihren Ball = subject + object
• sieht = finite verb at the end

This is a very important German pattern:

• Ich bleibe zu Hause, wenn es regnet.
• Sie freut sich, weil er kommt.
• Er sagt, dass er müde ist.

In all of these, the verb goes to the end of the subordinate clause.

Because Ball is the direct object of sieht, so it must be in the accusative.

Step by step:

• Ball is masculine: der Ball
• direct object of sehen takes the accusative
• masculine accusative changes from der to den
• possessive words change in a similar way

So:

• ihr Ball = her ball as a subject
• ihren Ball = her ball as a direct object

That is why the sentence has:

sie ihren Ball sieht

Also note: the possessive ihr- means her here because it refers to sie = she.

By itself, sie can mean several things:

• she
• they
• you formal, if written Sie

Here it clearly means she, because it refers back to die kleine Hündin, which is singular and feminine.

So in this sentence:

• die kleine Hündin = the little female dog
• sie = she

If it were formal you, it would normally be written with a capital letter: Sie.

Yes. A more straightforward version would be:

Die kleine Hündin wedelt im Garten noch stärker, wenn sie ihren Ball sieht.

This has a more typical learner-friendly order:

• subject first
• verb second
• place later

The original sentence is completely normal, though. It just puts Im Garten first for emphasis or flow.

Both are grammatical; the difference is mainly about focus, not basic meaning.