In this sentence, is the word "were" or "was"?

I agree with both, as it depends what you define as the subject.

If "Jarrod Scarbrough and Les Sewell" is the subject, then the conjugation is "were"

But if "the Hillsborough couple" if the subject, then the conjugation is "was".

You could probably save the confusion by re-writing the sentence to:

"After 19 solid years of withstanding the abrasive predicaments in their amorous union, Jarrod Scarbrough and Les Sewell, the Hillsborough couple, was denied the right to a marriage license in Tampa, Florida."

Then your teacher would have no quarrels with it.

I agree with your teacher, couple can sometimes be considered plural.

"If couple conveys the idea of two people, treat it as a plural: The couple [its members] were married. But: Each couple was asked to give $10."

There are differences in the English language from the US to the UK and this might be one of them, so depending on where you are and where these answers are coming from you might not be getting the right results. It would be best to ask your teacher to explain it to you. Then at least you can tell if they're sure about it or not.

You are right. The subject is couple, which is singular. You could leave out the names of the two people in the couple, the names which you have properly put between commas, and the sentence would read perfectly.

You are correct. The subject is the singular 'couple.' The names form an appositive, set off with commas, and an appositive cannot change thw wors with which it is in apposition.

I agree with you. If you leave out the two names, the answer is obvious. But, it looks funny. Stick the names somewhere else so you don't place them in this sentence.

Were would be correct cause its past tense