Do you prefer to stay home, or do you want to come with us?
Should I prepare dinner, or are we going to go to the restaurant?
Are the questions correctly written?
Is there a preferred way to write such questions?
What you're doing here is combining two questions or two independent clauses with a coordinating conjunction, which conjoins two or syntactically identical units. The comma before the conjunction (when dealing with two clauses) is stylistic and optional, though I'd personally recommend it.
Your examples look perfect.