Learn Danish: Du har ikke et alibi, vel? - You do not have an alibi, do you?

Because the main sentence is negative (it contains ikke). In Danish, when you add a tag question like 'right?' or 'do you?' to a negative statement, you must use vel. You only use ikke også after positive statements.

How would I change this sentence to say 'You have an alibi, don't you?'

You would remove ikke to make the statement positive, and change the tag question to ikke også. The sentence would be: Du har et alibi, ikke også?

Why does ikke come after har and not before it?

This is because of the Danish V2 rule. In a main clause, the verb (har) must be the second piece of information. Since the subject (Du) takes the first spot, the verb comes second, and the negative word (ikke) takes the third spot.

Is the word alibi treated differently in Danish than in English?

It means the same thing and is spelled exactly the same way! Grammatically, it is a neuter noun, which is why it takes et (et alibi). Just keep in mind the pronunciation is slightly different, with shorter vowel sounds than the English version.