Learn Hardcore Japanese: watasi no inu ha niwa de genki ni hasirimasu.

は: topic marker → As for my dog… (background information, what we’re talking about)
が: subject marker → often highlights or introduces the subject, or contrasts it with others

Possible uses of が here:

Answering a question like:
誰が庭で元気に走りますか。 (Who runs energetically in the garden?)
→ 私の犬が庭で元気に走ります。
(It is) my dog (that) runs energetically in the garden.
Contrasting:
猫は寝ていますが、私の犬が庭で元気に走ります。
The cat is sleeping, but my dog is running energetically in the garden.

If you’re just stating a fact about your dog with no contrast or emphasis, は is more natural:
私の犬は庭で元気に走ります。

What does the particle で (de) after 庭 mean? Why not に (ni)?

で marks the place where an action happens.

庭で = in the garden / at the garden (as the place where the action happens)

Basic difference:

で → location of an action / event
に → location of existence or destination / goal

Compare:

庭で犬が走ります。
The dog runs in the garden. (running is an action → で)
庭に犬がいます。
There is a dog in the garden. (existence / staying → に)
庭に行きます。
I will go to the garden. (destination → に)

In your sentence, 走ります (runs) is an action, so the place of that action takes で:
庭で走ります = runs in the garden.

I thought 元気 (genki) means “healthy” or “well.” What does 元気に mean in this sentence?

You’re right that 元気 often means “healthy / well / lively.”

In your sentence, 元気に is used as an adverb meaning something like:

energetically
lively
playfully / with lots of energy

So:

元気に走ります。
≈ runs energetically / runs around full of energy

Other common uses:

元気です。 – I’m fine / I’m well / I’m energetic.
子どもたちは元気です。 – The children are lively / full of energy.

Here you’re describing how the dog runs, so it becomes an adverb: 元気に.

Why is there に (ni) after 元気? How does 元気に work grammatically?

元気 is a na-adjective (sometimes called an adjectival noun).
Na-adjectives often form their adverb by adding に.

Pattern:

きれいな花 – a pretty flower
きれいに書く – write neatly / nicely
静かな公園 – a quiet park
静かに話す – speak quietly
元気な子ども – a lively child
元気に走る – run energetically

So 元気に is “元気 in an adverb form” → describing how the verb is done.

Is the word order fixed? Can I say 庭で私の犬は元気に走ります or 元気に私の犬は庭で走ります?

Japanese word order is fairly flexible, as long as the particles stay attached to the right words.

All of these are grammatically okay:

私の犬は庭で元気に走ります。 (original)
庭で私の犬は元気に走ります。
元気に私の犬は庭で走ります。 (a bit marked but possible)

However:

The verb almost always comes last: 走ります should stay at the end.
The most natural / neutral version here is probably the original:
- 私の犬は庭で元気に走ります。

Putting 庭で or 元気に first can slightly change the focus:

庭で私の犬は元気に走ります。
Puts a bit of emphasis on in the garden.
元気に私の犬は庭で走ります。
Sounds a bit literary / stylistic, emphasizing energetically.

For everyday use, stick with the original order while you’re learning.

Can I drop 私の and just say 犬は庭で元気に走ります?

Yes, you can. That sentence is perfectly natural:

犬は庭で元気に走ります。
The dog runs energetically in the garden.

In Japanese, pronouns and possessives are often omitted when the meaning is clear from context.

So in a conversation where it’s obvious you’re talking about your own dog, people will understand 犬は庭で元気に走ります as “My dog runs energetically in the garden.”

You include 私の if:

You need to clarify ownership
You’re contrasting with someone else’s dog:
田中さんの犬は静かですが、私の犬は庭で元気に走ります。
Mr. Tanaka’s dog is quiet, but my dog runs energetically in the garden.

Why is 走ります used instead of the dictionary form 走る? What does 〜ます add?

走る is the dictionary (plain) form.
走ります is the polite form.

In Japanese, verbs conjugate for politeness, not for person:

走る – plain form: used with friends, family, in casual writing, in dictionaries.
走ります – polite ます-form: used in most public / formal situations, with people you don’t know well, in textbooks, etc.

Same meaning (present / future), different politeness level:

私の犬は庭で元気に走る。 – casual
私の犬は庭で元気に走ります。 – polite

Textbooks usually teach the ます-form first because it’s safe in most situations.

How would I say “My dog ran energetically in the garden” (past tense)?

Just change 走ります to its past polite form 走りました:

私の犬は庭で元気に走りました。
My dog ran energetically in the garden.

Patterns for ます verbs:

走ります → 走りました (run → ran)
食べます → 食べました (eat → ate)
行きます → 行きました (go → went)

How would I say “My dog is running energetically in the garden right now”?

Use the te-form + います construction to show a continuous action:

Dictionary form: 走る
Te-form: 走って
Continuous polite: 走っています

So:

私の犬は庭で元気に走っています。
My dog is running energetically in the garden (now).

This pattern is used a lot:

食べています。 – I am eating.
読んでいます。 – I am reading.

Why is は pronounced “wa” in 犬は instead of “ha”?

The character は normally represents the sound “ha.”

However, when は is used as the topic particle, it is always pronounced “wa.”

So:

In words:
- はな → hana (flower/nose)
- はし → hashi (chopsticks/bridge)
As topic particle:
- 犬は → inu wa
- 私は学生です。 → watashi *wa gakusei desu.*

It is written は but read “wa” in that grammatical role. This is just a historical spelling convention.

Why are there spaces between the words in 私　の　犬　は　庭　で　元気　に　走ります? I thought Japanese doesn’t use spaces.

You’re right: normal Japanese writing does not use spaces between words.

A real sentence would usually look like this:

私の犬は庭で元気に走ります。

Spaces are often added in textbooks, beginner materials, or grammar explanations to help learners see:

where one word ends and another begins
which particle attaches to which word

So the spaced version is just for teaching:

私　の　犬　は　庭　で　元気　に　走ります。 (teaching version)
私の犬は庭で元気に走ります。 (normal Japanese)