Learn German: Der Herr und die Dame sind schon alt. - The gentleman and the lady are already old.

Because Der Herr is part of the subject here, so it is in the nominative case. N-declension (or weak) masculine nouns only add that extra -n or -en in the accusative, dative, and genitive cases. In the nominative singular, the noun stays in its base form: der Herr.

Why do we use the plural verb sind when both der Herr and die Dame are singular nouns?

Because they are joined by und (and) to form a single plural subject. Just like in English, the gentleman and the lady acts as a plural group (they), so the verb must be the plural sind.

Why doesn't the word alt have an adjective ending like alte or alten here?

Adjectives in German only get endings when they are placed directly in front of a noun (like der alte Herr). When an adjective comes after the verb to describe the subject, it is called a predicate adjective and it stays in its basic dictionary form.

I usually see Herr used as Mr. How does it mean gentleman here?

Herr is versatile! When used as a title before a name, it means Mr. (like Herr Schmidt). But when used as a standalone noun with an article like der, it translates to the gentleman.