Learn Hardcore Russian: Ей не нравится миска.

The verb нравиться always takes the experiencer (the one who likes or dislikes something) in the dative case. Она would be nominative, which doesn’t fit the pattern дative + нравиться + nominative.

How is the verb нравиться structured in terms of cases?

With нравиться, the thing being liked (or disliked) is in the nominative case and acts like the grammatical subject, while the person who likes it is in the dative case.

Why is миска in the nominative case and not in the accusative?

Because in the нравиться-construction the thing that “pleases” is considered the subject of the verb, so it stays in the nominative. The accusative is used with verbs like любить, but not with нравиться.

Where is the grammatical subject of the sentence?

Grammatically, миска is the subject (in nominative) and нравится is the verb. Semantically, though, ей is the one doing the “not liking.”

How would you say “I don’t like the bowl” in Russian?

You switch the dative pronoun to мне:
Мне не нравится миска.

What’s the difference between нравиться and любить?

Нравиться means “to be pleasing to” (more like “to like”) and uses dative + nominative. Любить means “to love” (or strongly like) and takes the object in the accusative:
Она не любит миску.

How would you ask “Does she like the bowl?” in Russian?

Colloquially you can just say
Ей нравится миска?
Formally (or to add emphasis) you can use the particle –ли:
Нравится ли ей миска?

Can you change the word order, and if so, does it affect meaning?

Yes. Russian is fairly flexible. You could say:
Миска ей не нравится.
or
Ей миска не нравится.
The basic meaning stays the same; word order shifts emphasis (e.g., starting with миска highlights the bowl).

Why is не placed before нравится rather than before миска?

Putting не before нравится negates the verb “doesn’t please.” If you moved не before миска:
Ей нравится не миска, а чашка,
you’d be contrasting objects (“It’s not the bowl she likes but the cup”), not simply negating the liking itself.