Learn Hardcore Icelandic: Bíllinn er þinn.

As a masculine noun, its definite ending is -inn → bíllinn.
The possessive pronoun must agree in gender (as well as number and case), so you use the masculine form þinn rather than þín or þitt.

Why is the possessive pronoun þinn used instead of þín or þitt?

Icelandic possessive pronouns change form to match the gender, number, and case of the noun they refer to:

þinn = masculine singular (matches bíll)
þín = feminine singular
þitt = neuter singular
Since bíll is masculine singular, we use þinn.

What case is þinn in here, and why is it nominative rather than genitive?

In Bíllinn er þinn, þinn is a predicate adjective/pronoun in the nominative case because it follows the copula er (“is”).

Predicate nouns and adjectives always take the nominative in Icelandic.
Genitive would be used for possession in a noun phrase (e.g. bíll barnsins “the child’s car”), but not after vera.

Why is the verb er necessary here? Could you omit it?

Er is the 3rd person singular present of vera (“to be”), and Icelandic requires a copula in equational sentences.

Bíllinn þinn by itself is just a noun phrase meaning “your car.”
Bíllinn er þinn is a full sentence: “The car is yours.” You cannot drop er without changing it from a sentence into a bare noun phrase.

How would you ask “Is the car yours?” in Icelandic?

You form a yes/no question by inverting subject and verb:
Er bíllinn þinn?
Literally: “Is the-car-the yours?”

How would you say “the car is his” or “the car is ours” in the same structure?

You replace þinn with the appropriate possessive pronoun (in nominative):

Can you also express “your car” as a simple noun phrase, and if so, how?

Yes. There are two common ways:

Preposed pronoun (no article suffix): þinn bíll (“your car” indefinite)
Postposed pronoun (with article suffix): bíllinn þinn (“your car” definite)
Both mean “your car,” but bíllinn þinn literally combines “the car” and “yours.”