Learn Hardcore Russian: Моя собака свободно бегает в саду.

собака is feminine, singular, and here it’s the subject, so it’s in the nominative case.
моя is the feminine singular nominative form of “my.”
мой would be masculine nominative, моё neuter nominative, and моей is either genitive or dative feminine, so they don’t fit here.

Why is свободно an adverb and not свободная?

свободная is an adjective, and adjectives modify nouns (e.g., свободная собака “a free dog”).
свободно is an adverb, and adverbs modify verbs, adjectives, or other adverbs. Here it describes how the dog runs, so we need the adverb свободно (“freely”).

What aspect and meaning does бегает convey, and how is it different from бежит?

бегает is the imperfective, present-tense form of бегать. Use it for habitual or repeated actions: “She runs (regularly).”
бежит is the imperfective, present-tense form of бежать, but it often emphasizes a one-time or in-progress action (“is running right now”).
Choosing бегает here suggests the dog runs freely in the garden as a general habit or ongoing situation, not necessarily right at this second.

Which case is в саду, and why does it end in -у instead of -е?

в саду is the prepositional case used after the preposition в to indicate location.
Some masculine nouns like лес, сад, луг have a special locative form ending in -у instead of the regular -е. So instead of в саде, standard usage is в саду.

Why do we say в саду and not на саду or just сад?

в + prepositional case indicates being inside an area or space (“in the garden”).
на typically goes with open surfaces (e.g., на крыше “on the roof”) or with places considered as points (e.g., на площади “on the square”).
Omitting the preposition isn’t allowed here, because you need в to show location inside the garden.

How would you express this sentence in the past and future tenses?

Past tense (imperfective, habitual or descriptive past):

Моя собака свободно бегала в саду.
(Here бегала is feminine singular past imperfective.)

Future tense (imperfective, describing a habitual or planned future action):

Моя собака будет свободно бегать в саду.
(Compound future with будет + infinitive for imperfective.)

If you want a single completed event in the future, you’d use the perfective:

Can you move свободно to a different position in the sentence for emphasis?

Yes, Russian word order is relatively flexible. All of these are grammatically correct, though the nuance shifts slightly:

Моя собака свободно бегает в саду. (Neutral statement)
Свободно моя собака бегает в саду. (Emphasizes freely)
В саду моя собака свободно бегает. (Emphasizes location)
Моя собака бегает свободно в саду. (Less common; adverbs usually sit close to the verb)

How do you express “is running” (present continuous) versus “runs” (simple present) in Russian?

Russian does not distinguish between simple and continuous present in form. The present tense of an imperfective verb covers both:

Моя собака бегает в саду. can mean either “My dog runs in the garden” (habitual) or “My dog is running in the garden” (right now), depending on context.
To stress the ongoing action you can add words like сейчас (“now”):
Моя собака сейчас свободно бегает в саду.