Learn Finnish: Viisi ja kaksi on seitsemän.

In Finnish math equations, the combination of numbers is treated as a single abstract idea. Because of this, you always use the third-person singular verb on (is), never the plural ovat (are).

Is ja the official word for "plus" in Finnish math?

ja simply means "and", but it is very commonly used for basic addition in everyday speech. If you want to use the strict mathematical term for "plus", you would use the borrowed word plus (making it viisi plus kaksi).

I learned that numbers usually make the following word take the partitive form. Why isn't there any partitive here?

That rule only applies when you are counting objects or things (like kaksi autoa for "two cars"). In this sentence, you are just talking about the numbers themselves, not counting anything. When numbers stand alone like this, they stay in their basic dictionary form.

Would a Finnish speaker say these exact words in a casual conversation?

In written Finnish, yes. However, in everyday spoken Finnish, numbers are almost always shortened. You will very likely hear viis instead of viisi, kaks instead of kaksi, and seittemän instead of seitsemän.