Learn Hardcore German: Der Apfel sieht frisch aus.

aussehen is a separable verb made up of the root sehen and the prefix aus-. In a main clause the conjugated part (sieht) occupies the second position, and the prefix (aus) moves to the end of the clause:
Der Apfel | sieht | frisch | aus.

Why is Der Apfel in the nominative case?

Because Der Apfel is the subject of the sentence. The verb aussehen does not take an object; it links a subject (in nominative) to a predicative adjective.

Why does frisch have no ending like frischer or frisches?

When an adjective is used predicatively (i.e. after a verb like aussehen), it remains in its base form and is not declined.

Could I replace frisch with another adjective, for example saftig?

Yes. Any adjective used predicatively stays uninflected. For example:
Der Apfel sieht saftig aus.

How would I form a yes/no question from this sentence?

Move the finite verb to the first position:
Sieht der Apfel frisch aus?

How can I express that the apple looks fresh to me personally?

Use the dative pronoun mir to indicate your impression:
Der Apfel sieht mir frisch aus.
Alternatively, you can say:
Der Apfel sieht für mich frisch aus.

Can I use scheinen or wirken instead of aussehen, and what’s the difference?

Yes, but with nuances:

aussehen refers strictly to visual appearance.
scheinen means “to seem” in a more general sense and often pairs with zu
- infinitive (Der Apfel scheint frisch zu sein).
wirken implies the effect or impression something has on someone (Der Apfel wirkt frisch).

Why is the verb always in the second position, even though Der Apfel is two words?

German main clauses follow the V2 (verb-second) rule: the finite verb must appear in the second position regardless of how many words form the first constituent.

How would the sentence look in a subordinate clause?

In subordinate clauses the prefix rejoins the verb at the end of the clause:
… dass der Apfel frisch aussieht.