The verb agrees with the first element in the set: There is a man and a woman in the room. Also: There is an apple, a banana and three oranges on the table. but There are three oranges, an apple and a banana on the table.

It depends on the noun. In most sentences, the noun comes before the verb. But in sentences that begin with there is and there are, the noun comes later. - There is a man. (Man is singular, so "there is) - There are two men. (Men is plural, so"there are") When you are making a list of things (e.g. a kitchen, a living room, a bedroom, ...), it sounds wrong to use "there are" when the noun is singular: a man, a woman. Even though there are more persons involved, it’s often better to use the singular verb in a construction like this. However, linguists might insist that "there are" is more correct, because multiple people are in the room. "They are in the room". Or if you turn around the sentence: A man and a woman are (not is) in the room. So both are possible, especially in spoken form.

There is a man and a woman in the room. It is the right structure. P.S. I can't explain grammatical rules. Because I never learned it. But it is the right way.