Learn Finnish: Miten tämä sanotaan ruotsiksi?

The ending -ksi is the Translative case. In Finnish, when you are talking about the language of communication—such as how something is spoken, written, or translated—you attach -ksi to the name of the language. So, ruotsi (Swedish) becomes ruotsiksi (in Swedish).

What verb form is sanotaan?

It is the present passive form of the verb sanoa (to say). You use the passive when the specific person doing the action isn't important. It translates perfectly to "how is [it] said" or "how does one say [it]".

"This" is the object being said, so why isn't tämä in an object case like the partitive or genitive?

In Finnish, when a sentence is in the passive voice (like sanotaan), a whole object doesn't take the normal -n ending (the genitive-accusative). Instead, it stays in the basic nominative form. That is why it remains just tämä.

Can I use kuinka instead of miten to mean "how" here?

Yes, you can! Kuinka tämä sanotaan ruotsiksi? means exactly the same thing. However, miten is extremely common in everyday spoken Finnish when asking for instructions or translations.