Pies is in the nominative singular, because it is the subject of the sentence.
The basic structure is:
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:
But in Mój pies nie jest już chory, the dog is the one being described, so pies is nominative.
In Polish, nie is usually written separately from finite verbs.
So:
This is normal Polish spelling:
In this sentence, już means any longer / anymore / now depending on how English expresses it.
So nie jest już chory means:
Literally, już often means already, but with negation it very often gives the sense of no longer:
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:
You may also hear other arrangements, such as:
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:
Other forms appear in other grammatical contexts:
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:
No. In this sentence, on is not needed.
Polish usually does not add a subject pronoun if the subject is already expressed:
Adding on would usually be unnecessary or emphatic:
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:
A rough pronunciation guide is:
mooy pyesh nye yest yoosh HO-rɨ
A few key points:
A more careful approximation:
Not in natural English. Even though już often means already, in a negative sentence like this the natural English meaning is:
So learners should be careful not to translate już word-for-word every time. Its meaning changes with context, especially after nie.