"Hispanice" technically isn't a word for the language Spanish. It's derived from the name "Hispania," which was the name for the general region of Spain in Roman times. "Hispanice" is the ablative form of the adjective "hispanicus" and so means something like "in a Hispania-ish manner."
So the sentence more literally says "They speak in a Hispania-ish manner," which we can in modern day understand to be "in Spanish." (But of course their language would not have been Spanish in Roman times.)
Neither of those because adverbs do not have cases. Latine is an adverb from the adjective Latinus, -a, -um meaning "latinly".
If you really wanted to use that phrase it could be: In lingua Latina loquimur.
"We speak in the Latin tongue."
In + ablative.
Or more biblically: In linguis loquimur.
"We speak in tongues."