Cura/curar is related from its Latin root to cuidar to care for or to look after. So it has taken a couple of different paths, one to cure or heal and the other for the person in the Parrish who is responsible to take care of the flock. As priest it is used especially to delineate the specfic responsibility of a particular priest in addition to a general meaning.
They mostly actually came into English from Norman French which was spoken in England as the language of law and culture (e.g. the upper classes) for a couple of hundred years after the Norman Invasion of 1066, although we did get some directly from Latin and a few from French. That is why, while it is hard for a modern English speaker to read Middle English, old English is next to impossible as it is much more Germanic. It makes it easier to learn Romance languages, although you do have to beware of false cognates.
Yes, at least in English here. In Germany not all physicians are doctors because they have to complete a doctorate to be a doctor. But in terms of Dúo, it would work a lot better and with fewer problems if there were only one word for a Spanish word. Obviously that isn't the case with many words, but when you have a clear true cognate that is what Dúo will use. It can be difficult for people learning from another second language, but if you are a native speaker of English you know to what extent a word has synonyms. It is not Duo's goal to allow many options for any word. They provide the meaning by choosing an English word that is equivalent, then you can find the synonyms yourself. Some words will still require multiple matches depending on the circumstances, but if you have a cognate that works, always choose it. In addition because of the cognate most people would translate doctor(a) as doctor and médico/a as physician
But recently, for example in Italy, if you are a doctor that cures patients, you have to include "DM" - "Dottore in Medicina" after your name (like "MD" - "Medical Doctor" in England/USA). This is not to create misunderstandings with the patients. The term "Doctor" is by now too generic.