Learn Italian: Il nostro papà ha sempre ragione.

You are right that singular family members usually drop the article, so you would say nostro padre (our father). However, affectionate names like papà (dad) and mamma (mom) are exceptions. They always keep the article, which is why it is il nostro papà.

Why does the sentence use "ha" (has) instead of "è" (is)?

In English, you are right, but in Italian, you have reason. The phrase is avere ragione (to have reason). Since we are talking about il nostro papà (he), we use the "he" form of the verb avere, which is ha.

Does "ragione" just mean "right"?

Literally, ragione means "reason". So ha sempre ragione translates word-for-word as "has always reason". Italians use this noun to mean someone is correct. If you mean "right" as a direction, you would use the word destra.

Where do I place "sempre" in a sentence like this?

Adverbs of frequency like sempre (always) usually go immediately after the conjugated verb. Here, it sits naturally right between the verb ha and the noun ragione.