"I do not learn Latin."
Translation:Linguam Latinam non disco.
"Latinam linguam non disco" is perfectly valid Latin. Placing "Latinam" first merely emphasises that word. I think a Roman would have understood that sentence to mean, "I do not learn the Latin language", i.e. as opposed to any other language. But it is still a grammatical sentence.
People are often confused about Latin word order. They think that just because there are very few hard and fast rules about where to place the words, it means that word order is random. It isn't!
Yes. As far as I remember, there is actually no word for 'the Latin language' in Latin. Latinus,-a,-um is an adjective so it needs to go with the noun 'lingua'. It is also sometimes used by itself to speak of the 'Latins' i.e, the ancient Italian ethnic group. You can, however, use the word 'Latine' which is indeclinable and always means 'in Latin', i.e. 'possum loqui Latine' meaning, 'I can speak in Latin'.