Learn German: Reinrassige Hunde sollen am allerschönsten sein. - Purebred dogs are said to be the very most beautiful.

The prefix aller- is added directly to a superlative adjective to intensify it, meaning "of all" or "the very." While am schönsten means "the most beautiful," am allerschönsten means "the very most beautiful" or "the most beautiful of all."

Why does sollen mean 'are said to be' here instead of 'should'?

At the B2 level, you'll often see modal verbs used to express hearsay or assumptions. When sollen is used this way (called the "subjective" use), it reports a claim made by someone else. Instead of expressing an obligation ("should"), it translates to "is said to be" or "is supposedly."

Could I say 'die allerschönsten sein' instead of 'am allerschönsten sein'?

Yes, but there is a subtle difference. Using am allerschönsten is the standard way to use a superlative to describe how something is (as a predicate adjective). If you said die allerschönsten, it acts more like a noun phrase, meaning "to be the most beautiful ones."

Why is the ending on Reinrassige just -e?

Because it modifies a plural noun (Hunde) with no article in front of it (no die, keine, meine, etc.). Without an article to show the grammatical case, the adjective has to do the heavy lifting and takes the "strong" declension ending. For nominative plural, that ending is -e.