Mój pies nie jest już chory.

Because pies (dog) is a masculine singular noun, and the possessive adjective mój has to agree with it in gender, number, and case.

• mój = masculine singular
• moja = feminine singular
• moje = neuter singular or non-masculine-personal plural

So:

• mój pies = my dog
• moja książka = my book
• moje dziecko = my child

Pies is in the nominative singular, because it is the subject of the sentence.

The basic structure is:

• Mój pies = subject
• nie jest = verb
• już chory = predicate/adjective phrase

So nothing is forcing pies into another case here.

Because pies is the subject, so it stays in the nominative.

Psa is a different case form, usually accusative or genitive singular:

• Widzę psa. = I see the dog.
• Nie ma psa. = There is no dog / The dog is not here.

But in Mój pies nie jest już chory, the dog is the one being described, so pies is nominative.

Because Polish uses być (to be) to link the subject with an adjective, just like English does in My dog is sick.

Here:

• jest = is
• chory = sick / ill

So:

• pies jest chory = the dog is sick

With negation:

• pies nie jest chory = the dog is not sick

In Polish, nie is usually written separately from finite verbs.

So:

• jest = is
• nie jest = is not

This is normal Polish spelling:

• nie mam = I do not have
• nie wiem = I do not know
• nie jest = is not

In this sentence, już means any longer / anymore / now depending on how English expresses it.

So nie jest już chory means:

• is no longer sick
• is not sick anymore

Literally, już often means already, but with negation it very often gives the sense of no longer:

• Jest już gotowy. = He is already ready.
• Nie jest już gotowy. = He is no longer ready.

So the meaning of już depends on whether the sentence is positive or negative.

Yes, Polish word order is fairly flexible, but Mój pies nie jest już chory is a very natural order.

This order clearly gives:

• nie jest już chory = is no longer sick

You may also hear other arrangements, such as:

• Mój pies już nie jest chory.

This also means My dog is no longer sick and is very natural too.

The difference is mostly about emphasis and style, not basic meaning.

Because chory is an adjective agreeing with pies, which is masculine singular nominative.

So:

• chory pies = a sick dog
• pies jest chory = the dog is sick

Other forms appear in other grammatical contexts:

• chorego psa = accusative/genitive masculine singular
• chora = feminine singular
• chore = neuter singular or non-masculine-personal plural
• choro = usually an adverb, meaning something like sickly or used in expressions such as Jest mi niedobrze / Jestem chory, not as the agreeing adjective for pies

Yes. That would mean The dog is no longer sick or Dog is no longer sick, depending on context.

Polish often leaves out possessives when they are not necessary, but including mój makes it clear that you mean my dog specifically.

So both are correct:

• Pies nie jest już chory. = The dog is no longer sick.
• Mój pies nie jest już chory. = My dog is no longer sick.

No. In this sentence, on is not needed.

Polish usually does not add a subject pronoun if the subject is already expressed:

• Mój pies nie jest już chory.

Adding on would usually be unnecessary or emphatic:

• Mój pies, on nie jest już chory.
  This sounds marked or contrastive, not neutral.

So the normal version is the one you were given.

It can absolutely be used for animals too. Chory means sick / ill, and it works naturally for a dog.

So:

• Mój pies jest chory. = My dog is sick.
• Kot jest chory. = The cat is sick.

A rough pronunciation guide is:

mooy pyesh nye yest yoosh HO-rɨ

A few key points:

• ó in mój sounds like u
• ie in pies is pronounced together, roughly like pye-
• nie sounds like nye
• j in Polish sounds like English y
• już sounds roughly like yoosh
• ch in chory is like a throaty h

A more careful approximation:

• Mój ≈ mooy
• pies ≈ pyes
• nie jest ≈ nye yest
• już ≈ yoosh
• chory ≈ HO-rɨ

Not in natural English. Even though już often means already, in a negative sentence like this the natural English meaning is:

• My dog is no longer sick
• My dog isn’t sick anymore

So learners should be careful not to translate już word-for-word every time. Its meaning changes with context, especially after nie.